MATH 1330
Precalculus
Exercise Sets and
Odd-Numbered Solutions
University of Houston Department of Mathematics
MATH 1330, Precalculus: Exercise Sets and Odd-Numbered Solutions
© 2011 University of Houston Department of Mathematics
MATH 1330 – Precalculus
Exercise Sets and Odd-Numbered Solutions
Table of Contents
University of Houston Department of Mathematics
CHAPTER 1 EXERCISES
Exercise Set 1.1: An Introduction to Functions ........................................................................ 1
Exercise Set 1.2: Functions and Graphs ................................................................................... 5
Exercise Set 1.3: Transformations of Graphs ........................................................................... 9
Exercise Set 1.4: Operations on Functions ............................................................................. 13
Exercise Set 1.5: Inverse Functions ........................................................................................ 16
CHAPTER 2 EXERCISES
Exercise Set 2.1: Linear and Quadratic Functions.................................................................. 19
Exercise Set 2.2: Polynomial Functions ................................................................................. 23
Exercise Set 2.3: Rational Functions ...................................................................................... 26
Exercise Set 2.4: Applications and Writing Functions ........................................................... 29
CHAPTER 3 EXERCISES
Exercise Set 3.1: Exponential Functions ................................................................................ 32
Exercise Set 3.2: Logarithmic Functions ................................................................................ 35
Exercise Set 3.3: Laws of Logarithms .................................................................................... 38
Exercise Set 3.4: Exponential and Logarithmic Equations and Inequalities .......................... 42
Exercise Set 3.5: Applications of Exponential Functions....................................................... 45
MATH 1330 Precalculus
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TABLE OF CONTENTS
CHAPTER 4 EXERCISES
Exercise Set 4.1: Special Right Triangles and Trigonometric Ratios..................................... 48
Exercise Set 4.2: Radians, Arc Length, and the Area of a Sector........................................... 53
Exercise Set 4.3: Unit Circle Trigonometry ........................................................................... 57
Exercise Set 4.4: Trigonometric Expressions and Identities .................................................. 62
CHAPTER 5 EXERCISES
Exercise Set 5.1: Trigonometric Functions of Real Numbers ................................................ 65
Exercise Set 5.2: Graphs of the Sine and Cosine Functions ................................................... 67
Exercise Set 5.3: Graphs of the Other Trigonometric Functions Tangent, Cotangent, Secant, and Cosecant ......................................................... 73
Exercise Set 5.4: Inverse Trigonometric Functions................................................................ 80
CHAPTER 6 EXERCISES
Exercise Set 6.1: Sum and Difference Formulas .................................................................... 89
Exercise Set 6.2: Double-Angle and Half-Angle Formulas.................................................... 94
Exercise Set 6.3: Solving Trigonometric Equations ............................................................... 99
CHAPTER 7 EXERCISES
Exercise Set 7.1: Solving Right Triangles ............................................................................ 102
Exercise Set 7.2: Area of a Triangle ..................................................................................... 105
Exercise Set 7.3: The Law of Sines and the Law of Cosines ............................................... 108
ii
University of Houston Department of Mathematics
CHAPTER 8 EXERCISES
Exercise Set 8.1: Parabolas ................................................................................................... 112
Exercise Set 8.2: Ellipses ...................................................................................................... 115
Exercise Set 8.3: Hyperbolas ................................................................................................ 119
APPENDICES EXERCISES
Exercise Set A.1: Factoring Polynomials ............................................................................. 122
Exercise Set A.2: Dividing Polynomials .............................................................................. 124
ODD-NUMBERED ANSWERS TO EXERCISE SETS
Chapter 1 Odd Answers ....................................................................................................... 126
Chapter 2 Odd Answers ....................................................................................................... 139
Chapter 3 Odd Answers ....................................................................................................... 152
Chapter 4 Odd Answers ....................................................................................................... 168
Chapter 5 Odd Answers ....................................................................................................... 187
Chapter 6 Odd Answers ....................................................................................................... 210
Chapter 7 Odd Answers ....................................................................................................... 221
Chapter 8 Odd Answers ....................................................................................................... 225
Appendix Odd Answers........................................................................................................ 238
MATH 1330 Precalculus
iii
MATH 1330 – Precalculus
Textbook and Online Resource Information
University of Houston Department of Mathematics
Math 1330 Printed Textbook:
The packet you are holding contains only the exercise sets and odd-numbered answers to the
exercise sets. The entire Math 1330 printed textbook, which contains the lessons for each section
as well as these exercise sets and odd-numbered answers, can be purchased at the University
Copy Center.
Math 1330 Online:
All materials found in this textbook can also be found online at:
http://online.math.uh.edu/Math1330/
Online Lectures:
In addition to the textbook material, the online site contains flash lectures pertaining to most of
the topics in the Precalculus course. These lectures simulate the classroom experience, with
audio of course instructors as they present the material on prepared lesson notes. The lectures are
useful in furthering understanding of the course material and can only be viewed online.
iv
University of Houston Department of Mathematics
Exercise Set 1.1: An Introduction to Functions
For each of the examples below, determine whether
the mapping makes sense within the context of the
given situation, and then state whether or not the
mapping represents a function.
1.
Erik conducts a science experiment and maps the
temperature outside his kitchen window at
various times during the morning.
57
9
62
10
7.
Divide by 7, then add 4
8.
Multiply by 2, then square
9.
Take the square root, then subtract 6
10. Add 4, square, then subtract 2
65
Temp. (oF)
Time
2.
Express each of the following rules in function
notation. (For example, “Subtract 3, then square”
would be written as f ( x ) = ( x − 3)2 .)
Dr. Kim counts the number of people in
attendance at various times during his lecture this
afternoon.
1
85
2
Find the domain of each of the following functions.
Then express your answer in interval notation.
11. f ( x) =
5
x−3
12. f ( x) =
x−6
x +1
87
3
13. g ( x) =
# of People
Time
State whether or not each of the following mappings
represents a function. If a mapping is a function, then
identify its domain and range.
14. f ( x) =
15. f ( x) =
3.
7
9
-3
0
5
4
16. g ( x) =
x−4
x2 − 9
3x + 1
x2 + 4
x2 + 6x + 5
x 2 − 11x + 28
3 x + 15
2
x + 8 x − 20
A
B
17. f (t ) = t
4.
-6
8
4
-7
A
B
9
18. h( x) = 3 x
19. g ( x) = 5 x
20. h(t ) = 4 t
5.
9
-2
-6
21. f ( x) = x − 5
1
A
6.
B
0
8
23. F ( x) =
3 − 2x
x+4
24. G ( x ) =
x −3
x−7
2
4
A
22. g ( x) = x + 7
B
MATH 1330 Precalculus
1
Exercise Set 1.1: An Introduction to Functions
25. f ( x) = 3 x − 5
35. (a)
f (t ) = t
(b) g (t ) = 9 − t
26. g ( x) = 3 x 2 + x − 6
27. h(t ) = 3
t 2 − 7t − 8
t +5
(c) h(t ) = 9 − t
36. (a)
f ( x) = x
(b) g ( x) = x + 1
28. f ( x) = 5
2x − 9
4x − 7
29. f (t ) = t 2 − 10t + 24
(c) h( x) = x + 1
37. (a)
f ( x) = x 2 + 3
(b) g ( x) = x 2 + 3 − 4
30. g (t ) = t 2 − 5t − 14
Find the domain and range of each of the following
functions. Express answers in interval notation.
31. (a)
f ( x) = x
(b) g ( x) = x − 6
(c) h( x) = 2 x 2 + 3 + 5
38. (a)
f (t ) = t 2 + 6
(b) g (t ) = t 2 + 6 + 7
(c) h(t ) = − 13 t 2 + 6 − 8
(c) h( x) = x − 6
(d) p( x) = x − 6 + 3
32. (a) f (t ) = 3 − t
39. g (t ) = 2 x − 7 + 5
40. h(t ) = 6 − t − 1
(b) g (t ) = 3 − t
(c) h(t ) = 3 − t
(d) p(t ) = 3 − t − 7
33. (a)
42. g ( x) = 4 8 − 3x + 2
f ( x) = x 2 − 4
(b) g ( x) = 4 − x 2
(c) h( x) = x 2 + 4
(d) p( x) = x 2 − 4
(e) q ( x) = 4 − x 2
(f) r ( x) = x 2 + 4
34. (a)
41. f ( x) = 3 5 x + 6 − 4
Find the domain and range of each of the following
functions. Express answers in interval notation.
(Hint: When finding the range, first solve for x.)
3
x+2
x+5
(b) g ( x ) =
x−2
43. (a)
f ( x) =
f (t ) = 25 + t 2
(b) g (t ) = t 2 − 25
(c) h(t ) = 25 − t 2
(d) p(t ) = 25 + t 2
4
x−3
5x − 2
(b) g ( x) =
x−3
44. (a)
f ( x) =
(e) q (t ) = t 2 − 25
(f) r (t ) = 25 − t 2
2
University of Houston Department of Mathematics
Exercise Set 1.1: An Introduction to Functions
3x 2 , if x < 0
53. If f ( x) = 4, if 0 ≤ x < 2 , find:
x + 5, if x ≥ 2
Evaluate the following.
45. If f ( x) = 5 x − 4 , find:
( )
f (3), f − 12 , f (a ), f (a + 3), f (a) + 3, f (a ) + f (3)
f (0), f (−6), f (2 ), f (1), f (4), f
(32 )
46. If f ( x) = 3 x + 1 , find:
( )
f (5), f (−8), f − 74 , f (t ) − 2, f (t − 2), f (t ) − f (2)
−4 x − 7, if x ≤ −1
54. If f ( x) = x 2 + 6, if - 1 < x < 3 , find:
− 7, if x ≥ 3
47. If g ( x) = x 2 − 3 x + 4 , find:
( )
f (0), f (−4), f (− 1), f (3), f (6), f − 53
( ), g ( x + 5), g ( ), g (3a), 3g (a)
g (0), g −
1
4
1
a
48. If h(t ) = t 2 + 2t − 5 , find:
h(1), h
(32 ), h(c + 6), − h( x), h(2 x), 2h( x)
Determine whether each of the following equations
defines y as a function of x. (Do not graph.)
55. 3 x − 5 y = 8
49. If f ( x) =
2+ x
, find:
x −3
f (−7), f (0), f
(35 ),
56. x + 3 = y 2
f (t ), f (t 2 − 3)
57. x 2 + y = 3
58. 3 x 4 − 2 xy = 5 x
x2
50. If f ( x) =
− x , find:
x+4
59. 7 x − y 4 = 5
( )
f (2), f (−5), f − 74 , f (3 p), f ( p 3 ) + 2
60. 3 x 2 + y 2 = 16
61. x3 y − 2 y = 6
2 x − 5, if x ≥ 4
, find:
3 − x 2 , if x < 4
51. If f ( x) =
62.
f (6), f (2), f (− 3), f (0), f ( 4), f
()
9
2
x 2 + 4 x, if x < −2
52. If f ( x) =
, find:
7 − 2 x, if x ≥ −2
( )
f (−5), f (3), f (0 ), f (1), f (−2), f − 10
3
MATH 1330 Precalculus
x − 3y = 5
63. x3 = 5 y
64. y 3 = −7 x
65.
y −2= x
66.
x + 3y = 4
3
Exercise Set 1.1: An Introduction to Functions
For each of the following problems:
(a) Find f ( x + h) .
(b) Find the difference quotient
f ( x + h) − f ( x )
.
h
(Assume that h ≠ 0 .)
67. f ( x) = 7 x − 4
68. f ( x) = 5 − 3 x
69. f ( x) = x 2 + 5 x − 2
70. f ( x) = x 2 − 3 x + 8
71. f ( x) = −8
72. f ( x) = 6
4
73. f ( x ) =
1
x
74. f ( x ) =
1
x−3
University of Houston Department of Mathematics
Exercise Set 1.2: Functions and Graphs
Determine whether or not each of the following graphs
represents a function.
9.
y
x
1.
y
x
10.
2.
y
y
x
x
3.
y
x
4.
y
x
5.
y
11.
{(1, 5),
12.
{(−3, 2),
(1, 2), (0, − 3), (2, 1), (−2, 1)}
13. Analyze the coordinates in each of the sets
above. Describe a method of determining
whether or not the set of points represents a
function without graphing the points.
y
x
14. Determine whether or not each set of points
represents a function without graphing the
points. Justify each answer.
(a) {(−7, 3), (3, − 7), (1, 5), (5, 1), (−2, 1)}
(b)
7.
(2, 4), (−3, 4), (2, − 1), (3, 6)}
Answer the following.
x
6.
For each set of points,
(a) Graph the set of points.
(b) Determine whether or not the set of points
represents a function. Justify your answer.
y
(c)
(d)
{(6, 3), (−4, 3), (2, 3), (−3, 3), (5, 3)}
{(3, 6), (3, − 4), (3, 2), (3, − 3), (3, 5)}
{(−2, − 5), (−5, 2), (2, 5), (5, − 2), (5, 2)}
x
8.
y
Continued on the next page...
x
MATH 1330 Precalculus
5
Exercise Set 1.2: Functions and Graphs
(c) Find the y-intercept(s) of the function.
Answer the following.
15. The graph of y = f (x) is shown below.
10 y
f
(d) Find the following function values:
g (−2); g (0); g (1); g (3); g (6)
(e) For what value(s) of x is g ( x) = −2 ?
8
(f) On what interval(s) is g increasing?
6
(g) On what interval(s) is g decreasing?
4
(h) What is the maximum value of the function?
(i) What is the minimum value of the function?
2
x
−6
−4
−2
2
4
6
8
−2
17. The graph of y = g (x) is shown below.
−4
y
6
(a) Find the domain of the function. Write your
answer in interval notation.
g
4
(b) Find the range of the function. Write your
answer in interval notation.
2
(c) Find the y-intercept(s) of the function.
x
(d) Find the following function values:
−2
2
4
6
f (−2); f (0); f (4); f (6)
−2
(e) For what value(s) of x is f ( x) = 9 ?
(f) On what interval(s) is f increasing?
(g) On what interval(s) is f decreasing?
(h) What is the maximum value of the function?
(i) What is the minimum value of the function?
−4
(a) Find the domain of the function. Write your
answer in interval notation.
(b) Find the range of the function. Write your
answer in interval notation.
(c) How many x-intercept(s) does the function
have?
16. The graph of y = g (x) is shown below.
y
g
6
(d) Find the following function values:
g (−2); g (0); g (2); g (4); g (6)
(e) Which is greater, g (−2) or g (3) ?
(f) On what interval(s) is g increasing?
4
(g) On what interval(s) is g decreasing?
2
x
−2
2
4
6
−2
Continued on the next page…
(a) Find the domain of the function. Write your
answer in interval notation.
(b) Find the range of the function. Write your
answer in interval notation.
6
University of Houston Department of Mathematics
Exercise Set 1.2: Functions and Graphs
18. The graph of y = f (x) is shown below.
25. f ( x) = x − 3
y
6
26. f ( x) = 5 − x
f
4
27. F ( x) = x 2 − 4 x
2
28. G ( x) = ( x − 3) 2 + 1
x
−4
−2
2
4
−2
(a) Find the domain of the function. Write your
answer in interval notation.
(b) Find the range of the function. Write your
answer in interval notation.
29. f ( x) = x3 + 1
30. f ( x) = x 4 − 16
31. g ( x) =
12
x
32. h( x) = −
8
x
(c) Find the x-intercept(s) of the function.
(d) Find the following function values:
f (−3); f (−2); f (−1); f (1); f (4)
(e) Which is smaller, f (0) or f (3) ?
(f) On what interval(s) is f increasing?
(g) On what interval(s) is f decreasing?
For each of the following functions:
(c) State the domain of the function. Write your
answer in interval notation.
(d) Find the intercepts of the function.
(e) Choose x-values corresponding to the domain
of the function, calculate the corresponding yvalues, plot the points, and draw the graph of
the function.
3
2
19. f ( x) = − x + 6
20. f ( x) = 23 x − 4
21. h( x) = 3 x − 5, − 1 ≤ x < 3
22. h( x) = −2 x, − 3 < x ≤ 2
23. g ( x) = x − 3
For each of the following piecewise-defined functions:
(a) State the domain of the function. Write your
answer in interval notation.
(b) Find the y-intercept of the function.
(c) Choose x-values corresponding to the domain
of the function, calculate the corresponding yvalues, plot the points, and draw the graph of
the function.
2 x + 4, if − 2 ≤ x < 1
− x + 3, if 1 ≤ x ≤ 5
33. f ( x) =
1 x + 2, if − 3 ≤ x ≤ 0
34. f ( x) = 3
− 4 x + 3, if x > 0
3, if x < −2
− 5, if x ≥ −2
35. f ( x) =
−4, if − 5 ≤ x < 1
2, if 1 ≤ x ≤ 3
36. f ( x) =
4, if x < 0
2
x + 1, if x ≥ 0
37. f ( x) =
3 − x 2 , if x ≤ 1
− 3, if x > 1
38. f ( x) =
24. g ( x) = x − 4
MATH 1330 Precalculus
7
Exercise Set 1.2: Functions and Graphs
x, if x ≤ −3
39. f ( x) = x 2 , if − 3 < x < 2
4, if x ≥ 2
x 2 − 5, if x < 0
40. f ( x) = 1, if 0 ≤ x ≤ 3
2 − x, if x > 3
Answer the following.
41. (a) If a function is odd, then it is symmetric
with respect to the __________.
(x-axis, y-axis, or origin?)
(b) If a function is even, then it is symmetric
with respect to the __________.
(x-axis, y-axis, or origin?)
42. (a) If a function is symmetric with respect to the
y-axis, then the function is __________.
(Odd, even, both, or neither?)
(b) If a function is symmetric with respect to the
origin, then the function is __________.
(Odd, even, both or neither?)
43. Suppose that y = f (x) is an odd function and that
(−3, 6) is a point on the graph of f. Find another
point on the graph.
44. Suppose that y = f (x) is an even function and
that (2, − 7) is a point on the graph of f. Find
another point on the graph.
Determine whether each of the following functions is
even, odd, both or neither.
45. f ( x) = x3 − 5 x
46. f ( x) = x 2 + 3 x
47. f ( x) = x 4 + 2 x 2
48. f ( x) = x 5 + 2 x3
49. f ( x) = 2 x 3 + x 2 − 5 x + 1
50. f ( x) = 3 x 6 +
8
2
x2
University of Houston Department of Mathematics
Exercise Set 1.3: Transformations of Graphs
(a) Who is correct, Jack or Jill?
(b) Analyze the two methods and explain the
algebraic difference between the two. Use this
analysis to justify your answer in part (a).
Suppose that a student looks at a transformation of
y = f ( x ) and breaks it into the following steps. State the
transformation that occurs in each step below.
1.
y = 2 f (− x − 3) + 4
(a) From: y = f ( x )
To: y = f ( x − 3)
(b) From: y = f ( x − 3)
To: y = f (− x − 3)
4.
y = x 2 , Fred first reflects the graph in the y-axis and
then shifts four units to the left. Wilma, on the other
hand, first shifts the graph y = x 2 four units to the
left and then reflects in the y-axis. Their graphs are
shown below.
(c) From: y = f (− x − 3) To: y = 2 f (− x − 3)
(d) From: y = 2 f (− x − 3)
To: y = 2 f (− x − 3) + 4
2.
Fred and Wilma are graphing the
function f ( x) = (− x + 4)2 . Starting with the graph of
y = − 14 f (1 − x) − 5
Fred’s Graph:
First notice that (1 − x) is equivalent to (− x + 1)
(a) From:
To:
(b) From:
(c) From:
(d) From:
y=
1
4
y=
1
4
f (1 − x)
5.
y = − f (1 − x)
y = − 14 f (1 − x) − 5
To:
Answer the following.
3.
Jack’s Graph:
Jill’s Graph:
y
y
2
2
x
−2
2
x
−6 −4 −2
2
−2
−2
−4
−4
−6
−6
4
6
Tony and Maria are graphing the
function f ( x) = −2 − x . Starting with the graph of
Tony’s Graph:
Jack and Jill are graphing the function
f ( x) = 2 − x 2 . Starting with the graph of y = x 2 ,
Jack first reflects the graph in the x-axis and then
shifts upward two units. Jill, on the other hand, first
shifts the graph y = x 2 upward two units and then
reflects in the x-axis. Their graphs are shown below.
−4
6
y = x , Tony first shifts the graph two units to the
right and then reflects in the y-axis. Maria, on the
other hand, first reflects the graph y = x in the yaxis and then shifts two units to the right. Their
graphs are shown below.
y = − 14 f (1 − x)
(e) From:
4
f (1 − x)
1
4
To:
2
x
2
y
(a) Who is correct, Fred or Wilma?
(b) Analyze the two methods and explain the
algebraic difference between the two. Use this
analysis to justify your answer in part (a).
y = f (1 − x )
To:
4
−6 −4 −2
y = f ( x + 1)
y = f (− x + 1) = f (1 − x)
To:
6
4
2
y = f ( x)
y = f ( x + 1)
Wilma’s Graph:
y
6
4
x
−4
−2
2
−2
−2
−4
−4
−6
−6
MATH 1330 Precalculus
Maria’s Graph:
6y
6
4
4
2
−6 −4 −2
2
x
2
4
6
y
−6 −4 −2
x
2
−2
−2
−4
−4
−6
−6
4
6
(a) Who is correct, Tony or Maria?
(b) Analyze the two methods and explain the
algebraic difference between the two. Use this
analysis to justify your answer in part (a).
4
9
Exercise Set 1.3: Transformations of Graphs
6.
Bart and Lisa are graphing the function
Write the equation that results when the following
f ( x) = 2 x − 3 . Starting with the graph of y = x , transformations are applied to the given standard
function. Then state if any of the resulting functions in
Bart first shifts the graph downward three units and
(a)-(e) are equivalent.
then stretches the graph vertically by a factor of 2.
Lisa, on the other hand, first stretches the graph
19. Standard function: y = x3
y = x vertically by a factor of 2 and then shifts the
graph downward three units.
Bart’s Graph:
Lisa’s Graph:
6y
6
4
4
2
2
y
x
x
−6 −4 −2
2
4
(a) Shift right 7 units, then reflect in the x-axis,
then stretch vertically by a factor of 5, then shift
upward 1 unit.
6
−6 −4 −2
2
−2
−2
−4
−4
−6
−6
4
6
(a) Who is correct, Bart or Lisa?
(b) Analyze the two methods and explain the
algebraic difference between the two. Use this
analysis to justify your answer in part (a).
Matching. The left-hand column contains equations that
represent transformations of f ( x ) = x 2 . Match the
equations on the left with the description on the right of
how to obtain the graph of g from the graph of f .
7.
g ( x) = ( x − 4)2
8.
g ( x) = x 2 − 4
9.
g ( x) = x 2 + 4
10. g ( x) = ( x + 4)2
11. g ( x) = − x 2
12. g ( x) = (− x)
13. g ( x) = 4 x
A. Reflect in the x-axis.
B. Shift left 4 units, then
reflect in the y-axis.
C. Reflect in the x-axis,
then shift downward 4
units.
D. Shift right 4 units.
2
2
E. Shift right 3 units, then
reflect in the x-axis, then
shift upward 4 units.
14. g ( x) = 14 x 2
F. Shift upward 4 units.
15. g ( x) = − x 2 − 4
H. Shift left 4 units, then
shift upward 3 units.
16. g ( x) = ( x + 4) 2 + 3
G. Reflect in the y-axis.
I.
Shift left 4 units.
17. g ( x) = −( x − 3) 2 + 4
J.
Shift downward 4 units.
18. g ( x) = (− x + 4) 2
K. Stretch vertically by a
factor of 4.
L. Shrink vertically by a
factor of 14 .
10
(b) Reflect in the x-axis, then shift right 7 units,
then stretch vertically by a factor of 5, then shift
upward 1 unit.
(c) Stretch vertically by a factor of 5, then shift
upward 1 unit, then shift right 7 units, then
reflect in the x-axis.
(d) Shift right 7 units, then shift upward 1 unit, then
reflect in the x-axis, then stretch vertically by a
factor of 5.
(e) Reflect in the x-axis, then shift left 7 units, then
stretch vertically by a factor of 5, then shift
upward 1 unit.
(f) Which, if any, of the resulting functions in (a)(e) are equivalent?
20. Standard function: y = x
(a) Reflect in the y-axis, then shift left 2 units, then
shift downward 4 units, then reflect in the xaxis
(b) Shift left 2 units, then reflect in the y-axis, then
reflect in the x-axis, then shift downward 4
units.
(c) Reflect in the y-axis, then reflect in the x-axis,
then shift downward 4 units, then shift right 2
units.
(d) Reflect in the x-axis, then shift left 2 units, then
shift downward 4 units, then reflect in the yaxis.
(e) Shift downward 4 units, then shift left 2 units,
then reflect in the y-axis, then reflect in the xaxis.
(f) Which, if any, of the resulting functions in (a)(e) are equivalent?
Continued on the next page…
University of Houston Department of Mathematics
Exercise Set 1.3: Transformations of Graphs
Describe how the graph of g is obtained from the graph of
f. (Do not sketch the graph.)
21. f ( x) = x ,
g ( x) = − x − 2
22. f ( x) = x3 ,
g ( x) = −2( x + 5)3
23. f ( x) = x ,
g ( x ) = −5 x − 2 + 1
24. f ( x) = x 2 ,
g ( x) = 16 ( x + 3) 2 − 7
1
25. f ( x) = ,
x
3
g ( x) =
+2
x +8
26. f (x ) = 3 x ,
g ( x) = 3 − x + 4
38. f ( x) = 2 x − 7
39. f ( x) = x 2 + 3
40. f ( x) = ( x − 5)2
41. f ( x) = 6 − x 2
42. f ( x) = 2 − ( x − 1) 2
43. f ( x) = −3 ( x − 4) 2 − 2
44. f ( x) = ( x + 5)2 + 3
45. f ( x) = 6 − x + 2
46. f ( x) =
Describe how the graphs of each of the following
functions can be obtained from the graph of y = f(x).
1
2
− x +1
47. f ( x) = − x + 4 + 2
27. y = f ( x) + 1
48. f ( x) = 5 − x − 1
28. y = f ( x − 7)
49. f ( x) = 2 x + 5 − 3
29. y = f (− x) + 3
30. y = − f ( x + 3) − 8
50. f ( x) = − x − 2 + 4
31. y = − 14 f ( x − 2) − 5
51. f ( x) = −( x − 4)3 + 1
32. y = −5 f (− x) + 1
52. f ( x) = − x3 − 5
33. y = f (7 − x) + 2
53. f ( x) =
1
+6
x−3
34. y = f (− x − 5) − 7
54. f ( x) = −
Sketch the graph of each of the following functions. Do
not plot points, but instead apply transformations to the
graph of a standard function.
2
x+4
4
x
55. f ( x) = − + 3
56. f ( x) = 3 x − 6
35. f ( x) = x − 3
36. f ( x) = 5 − x
37. f ( x) = −3 x + 1
MATH 1330 Precalculus
57. f ( x) = 3 − x + 2
58. f ( x) = − 3 x + 1 − 5
11
Exercise Set 1.3: Transformations of Graphs
The graph of y = f(x) is given below. Sketch the graph of
each of the following functions.
Answer the following.
59. The graphs of f ( x) = 4 − x 2 and g ( x) = 4 − x 2 are
10
shown below. Describe how the graph of g was
obtained from the graph of f.
y
8
6
y
f ( x) = 4 − x 2
y
g ( x) = 4 − x 2
4
2
2
2
x
−4
−2
2
f
4
4
x
4
−4
−2
−2
2
x
4
−8
−2
−6
−4
−2
2
4
6
8
10
−2
−4
−4
−4
−6
60. The graphs of f ( x) = x3 − 1 and g ( x) = x 3 − 1 are
−8
shown below. Describe how the graph of g was
obtained from the graph of f.
4
y
4
g ( x) = x3 − 1
f ( x) = x3 − 1
63. y = f ( x + 2)
y
2
2
64. y = f ( x) − 3
x
x
−4
−2
2
4
−4
−2
2
−2
−2
−4
−4
4
65. y = f ( x − 2) − 1
66. y = f ( x + 1) + 5
67. y = f (− x)
Sketch the graphs of the following functions:
61. (a) f ( x) = x 2 − 9
62. (a)
1
f ( x) =
x
(b) g ( x) = x 2 − 9
1
(b) g ( x ) =
x
68. y = − f (x)
69. y = 2 f ( x)
70. y = 12 f ( x)
71. y = −2 f ( x + 1)
72. y = f (− x) − 4
Continued in the next column…
12
University of Houston Department of Mathematics
Exercise Set 1.4: Operations on Functions
Answer the following.
1.
8
For each of the following problems:
(f) Find f + g and its domain.
y
g
(g) Find f − g and its domain.
6
f
4
(h) Find fg and its domain.
2
x
−8
−6
−4
−2
2
4
6
8
(i) Find
f
and its domain.
g
−2
Note for (a)-(d): Do not sketch any graphs.
−4
(a) Find f (−3) + g (−3) .
3.
f ( x) = 2 x + 3; g ( x) = x 2 − 4 x − 12
4.
f ( x) = 2 x3 − 5 x; g ( x) = x 2 + 8 x + 15
5.
f ( x) =
3
x
; g ( x) =
x −1
x+2
(f) Sketch the graph of f + g . (Hint: For any x
value, add the y values of f and g.)
6.
f ( x) =
4
2x
; g ( x) =
x−5
x −5
(g) What is the domain of f + g ? Explain how
you obtained your answer.
7.
f ( x) = x − 6 ; g ( x) = 10 − x
8.
f ( x) = 2 x − 3 ; g ( x) = x + 4
9.
f ( x) = x 2 − 9 ; g ( x) = x 2 + 4
(b) Find f (0) + g (0) .
(c) Find f (−6) + g (−6) .
(d) Find f (5) + g (5) .
(e) Find f (7) + g (7) .
8
2.
g
6
f
4
2
x
−8
−6
−4
−2
2
4
6
10. f ( x) = 49 − x 2 ; g ( x) = x − 3
8
−2
−4
Find the domain of each of the following functions.
−6
−8
11. f ( x) =
2
+ x −1
x−3
(a) Find f (−2) − g (−2) .
(b) Find f (0) − g (0) .
12. h( x) = x + 2 −
3
x
(c) Find f (−4) − g (−4) .
13. g ( x) =
3
x +1
−
x−7 x−2
(f) Sketch the graph of f − g . (Hint: For any x
value, subtract the y values of f and g.)
14. f ( x) =
x−2
5
+
+7
x + 6 x −1
(g) What is the domain of f − g ? Explain how
you obtained your answer.
15. f ( x) =
(d) Find f (2) − g ( 2) .
(e) Find f (4) − g ( 4) .
16. g ( x) =
x+2
x −5
x−3
x −1
MATH 1330 Precalculus
13
Exercise Set 1.4: Operations on Functions
Answer the following, using the graph below.
8
y
(a) Find the domain of g.
(b) Find f g .
g
6
f
4
2
x
−8
−6
−4
The following method can be used to find the domain
of f g :
−2
2
4
6
8
−2
(c) Look at the answer from part (b) as a standalone function (ignoring the fact that it is a
composition of functions) and find its domain.
(d) Take the intersection of the domains found in
steps (a) and (c). This is the domain of f g .
−4
17. (a) g (2)
(c) f (2)
18. (a) g (0)
(c) f (0)
(b) f (g (2 ))
(d) g ( f (2 ))
(b) f (g (0 ))
(d) g ( f (0 ))
19. (a)
( f g )(− 3)
(b)
(g f )(− 3)
20. (a)
( f g )(− 1)
(b)
(g f )(− 1)
21. (a)
( f f )(3)
(b)
(g g )(− 2)
22. (a)
( f f )(5)
(b)
(g g )(− 3)
23. (a)
( f g )(4)
(b)
(g f )(4)
24. (a)
( f g )(− 5) (b) ( f g )(2)
Note:
We check the domain of g because it is the inner
function of f g , i.e. f ( g ( x ) ) . If an x-value is
not in the domain of g, then it also can not be an
input value for f g .
Use the above steps to find the domain of f g for the
following problems:
33. f ( x) =
1
; g ( x) = x − 5
x2
34. f ( x) =
1
; g ( x) = x + 2
x2
35. f ( x) =
3
; g ( x) = x − 6
x −4
36. f ( x) =
5
; g ( x) = 3 − x
x −2
Use the functions f and g given below to evaluate the
following expressions:
2
f ( x) = 3 − 2 x and g ( x) = x − 5 x + 4
25. (a) g (0)
(c) f (0)
26. (a) g (−1)
(c) f (−1)
(b) f (g (0 ))
(d) g ( f (0 ))
(b) f (g (− 1))
(d) g ( f (− 1))
27. (a)
( f g )(− 2) (b) (g f )(− 2)
28. (a)
( f g )(4)
(b)
(g f )(4)
29. (a)
( f f )(6)
(b)
(g g )(6)
30. (a)
( f f )(− 4) (b) (g g )(− 4)
31. (a)
( f g )(x )
(b)
(g f )(x )
32. (a)
( f f )(x )
(b)
(g g )(x )
2
2
For each of the following problems:
(a) Find f g and its domain.
(b) Find g f and its domain.
37. f ( x) = x 2 + 3 x; g ( x ) = 2 x − 7
38. f ( x) = 6 x + 2; g ( x) = 7 − x 2
39. f ( x) = x 2 ; g ( x) =
1
x−4
14
40. f ( x) =
3
; g ( x) = x 2
x+5
University of Houston Department of Mathematics
Exercise Set 1.4: Operations on Functions
49. Given the functions
41. f ( x) = x + 7 ; g ( x) = −5 − x
f ( x) = x 2 + 4, g ( x ) = x + 3 , and h( x) = 2 x + 1, find:
(a) h( f (g (4 )))
(c) f g h
42. f ( x) = 3 − x ; g ( x) = 9 − 2 x
50. Given the functions
Answer the following.
f ( x) =
43. Given the functions f ( x) = x 2 + 2 and
g ( x) = 5 x − 8 , find:
(a)
(c)
(e)
(g)
f (g (1))
f (g (x ))
f ( f (1))
f ( f (x ))
(b) f (g (h(0 )))
(d) h f g
(b)
(d)
(f)
(h)
g ( f (1))
g ( f (x ))
g (g (1))
g (g (x ))
1
x2
, g ( x) = x − 2 , and h( x) = 3 − 4 x, find:
(a) h( f (g (5)))
(c) f g h
(b) f (g (h(− 2 )))
(d) h f g
Functions f and g are defined as shown in the table
below.
44. Given the functions f ( x) = x + 1 and
x
f ( x)
g( x )
g ( x) = 3 x − 2 x 2 , find:
(a)
(c)
(e)
(g)
f (g (− 3))
f (g (x ))
f ( f (− 3))
f ( f (x ))
(b)
(d)
(f)
(h)
45. Given the functions f ( x) =
g ( x) =
g ( f (− 3))
g ( f (x ))
g (g (− 3))
g (g (x ))
x +1
and
x−2
0
2
4
1
4
5
2
4
0
4
7
1
Use the information above to complete the following
tables. (Some answers may be undefined.)
51.
x
0
1
2
4
0
1
2
4
0
1
2
4
0
1
2
4
f ( g ( x ))
3
, find:
x −5
(a) f (g (− 2 ))
(c) f (g (x ))
(b) g ( f (− 2 ))
(d) g ( f (x ))
46. Given the functions f ( x) =
2x
and
x+5
52.
g ( f ( x ))
53.
7− x
g ( x) =
, find:
x −1
(a) f (g (3))
(c) f (g (x ))
x
x
f ( f ( x ))
(b) g ( f (3))
(d) g ( f (x ))
54.
x
g ( g ( x))
47. Given the functions
f ( x) = x 2 − 1, g ( x) = 3 x − 5, and h( x ) = 1 − 2 x, find:
(a) f (g (h(2 )))
(c) f g h
(b) g (h( f (3)))
(d) g h f
48. Given the functions
f ( x) = 2 x 2 + 3, g ( x) = x + 4, and h( x ) = 3x − 2, find:
(a) f (g (h(1)))
(c) f g h
MATH 1330 Precalculus
(b) g (h( f (1)))
(d) g h f
15
Exercise Set 1.5: Inverse Functions
Determine whether each of the following graphs
represents a one-to-one function. Explain your answer.
12. h ( x ) =
y
1.
11. g ( x ) = x + 4
x
1
−3
x
2
13. f ( x ) = − ( x − 2 ) + 1
14. g ( x ) = x − 6
2.
y
Answer the following.
x
15. If a function f is one-to-one, then the inverse
function, f −1 , can be graphed by either of the
following methods:
y
3.
(a) Interchange the ____ and ____ values.
(b) Reflect the graph of f over the line
x
y = ____ .
16. The domain of f is equal to the __________ of
f −1 , and the range of f is equal to the
y
4.
__________ of f −1.
x
A table of values for a one-to-one function y = f ( x ) is
given. Complete the table for y = f −1 ( x ) .
y
5.
17.
f −1 ( x )
x
3
f ( x)
−4
2
7
−4
5
5
5
0
0
0
3
x
5
f ( x)
9
x
4
5
5
6
−3
6
−8
2
−8
2
6
x
−4
x
6.
y
2
x
18.
For each of the following functions, sketch a graph
and then determine whether the function is one-to-one.
7.
f ( x) = 2x − 3
8.
g ( x ) = x2 + 5
9.
h ( x ) = ( x − 2)
f −1 ( x )
5
3
10. f ( x ) = x3 − 2
16
University of Houston Department of Mathematics
Exercise Set 1.5: Inverse Functions
For each of the following graphs:
(a) State the domain and range of f .
(b) Sketch f −1 .
Answer the following. Assume that f is a one-to-one
function.
23. If f ( 4 ) = 5, find f −1 ( 5) .
(c) State the domain and range of f −1 .
24. If f ( 6 ) = −2, find f −1 ( −2 ) .
19.
4 y
25. If f −1 ( −3) = 7, find f ( 7 ) .
2
x
−6
−4
−2
2
4
6
26. If f −1 ( 6 ) = −8, find f ( −8) .
f
−2
−4
27. If f ( 3) = 9 and f (9) = 5, find f −1 ( 9 ) .
−6
28. If f ( 5) = −4 and f (2) = 5, find f −1 ( 5) .
−8
29. If f ( −4 ) = 2, find f ( f −1 ( 2 ) ) .
20.
12
y
30. If f −1 ( −5) = 3, find f −1 ( f ( 3) ) .
10
8
f
6
Answer the following. Assume that f and g are defined
for all real numbers.
4
2
x
−4
−2
2
4
6
8
31. If f and g are inverse functions, f ( −2 ) = 3 and
−2
f ( 4 ) = −2 , find g ( −2 ) .
32. If f and g are inverse functions, f ( 7 ) = 10 and
21.
f (10 ) = −1 , find g (10 ) .
y
8
33. If f and g are inverse functions, f ( 5 ) = 8 and
f
6
f ( 9 ) = 3 , find g ( f ( 3) ) .
4
34. If f and g are inverse functions, f ( −1) = 6 and
2
x
−2
2
4
6
8
10
f ( 7 ) = 8 , find f ( g ( 6 ) ) .
−2
For each of the following functions, write an equation
for the inverse function y = f −1 ( x ) .
22.
8 y
f
35. f ( x ) = 5 x − 3
6
4
36. f ( x ) = −4 x + 7
2
x
−4
−2
2
4
−2
6
8
37. f ( x ) =
3 − 2x
8
38. f ( x ) =
6x − 5
4
−4
MATH 1330 Precalculus
17
Exercise Set 1.5: Inverse Functions
Answer the following.
39. f ( x ) = x 2 + 1 , where x ≥ 0
57. If f ( x ) is a function that represents the amount
40. f ( x ) = 5 − x 2 , where x ≥ 0
of revenue (in dollars) by selling x tickets, then
what does f −1 ( 500 ) represent?
41. f ( x ) = 4 x 3 − 7
58. If f ( x ) is a function that represents the area of a
42. f ( x ) = 2 x3 + 1
43. f ( x ) =
3
x+2
44. f ( x ) =
5
7−x
45. f ( x ) =
2x + 3
x−4
circle with radius x, then what does f −1 ( 80 )
represent?
A function is said to be one-to-one provided that the
following holds for all x1 and x2 in the domain of f :
If f ( x1 ) = f ( x2 ) , then x1 = x2 .
Use the above definition to determine whether or not
the following functions are one-to-one. If f is not oneto-one, then give a specific example showing that the
condition f ( x1 ) = f ( x2 ) fails to imply that x1 = x2 .
3 − 8x
46. f ( x ) =
x+5
47. f ( x ) = 7 − 2 x
59. f ( x ) = 5 x − 3
48. f ( x ) = 2 + 6 x + 5
60. f ( x ) = x3 + 5
Use the Property of Inverse Functions to determine
whether each of the following pairs of functions are
inverses of each other. Explain your answer.
49. f ( x ) = 4 x − 1 ;
g ( x ) = 14 x + 1
50. f ( x ) = 2 + 3 x ;
x−2
g ( x) =
3
61. f ( x ) = x − 4
62. f ( x ) = x − 4
63. f ( x ) = x − 4
64. f ( x ) =
51. f ( x ) =
4−x
;
5
g ( x ) = 4 − 5x
65. f ( x ) = x 2 + 3
1
2x + 5
52. f ( x ) = 2 x + 5 ;
g ( x) =
53. f ( x ) = x3 − 2 ;
g ( x) = 3 x + 2
54. f ( x ) = 5 x − 7 ;
g ( x) = ( x + 7)
55. f ( x ) =
5
;
x
1
+4
x
g ( x) =
66. f ( x ) = ( x + 3)
2
5
5
x
56. f ( x ) = x 2 + 9 , where x ≥ 0 ;
g ( x) = x − 9
18
University of Houston Department of Mathematics
Exercise Set 2.1: Linear and Quadratic Functions
Find the slope of the line that passes through the
following points. If it is undefined, state ‘Undefined.’
1.
(−2, 3) and (6, − 7)
2.
(−1, − 6) and (−5, 10)
3.
(8, − 7) and (−1, − 7)
4.
(3, − 8) and (3, − 4)
(a)
(b)
(c)
(d)
(e)
Write the equation in slope-intercept form.
Write the equation as a linear function.
Identify the slope.
Identify the y-intercept.
Draw the graph.
11. 2 x + y = 5
12. 3 x − y = −6
13. x + 4 y = 0
Find the slope of each of the following lines.
e
5.
14. 2 x + 5 y = 10
15. 4 x − 3 y + 9 = 0
y
c
4
c
6.
16. − 23 x + 12 y = −1
d
2
7.
e
x
8.
−4
f
−2
2
Find the linear function f that satisfies the given
conditions.
4
17. Slope −2
f
−4
18. Slope − 4 ; y-intercept 5
d
19. Slope −
Find the linear function f which corresponds to each
graph shown below.
4
; y-intercept 3
7
20. Slope
2
; passes through (-3, 2)
9
1
; passes through (-4, -2)
5
y
9.
21. Passes through (2, 11) and (-3, 1)
2
x
−4
−2
2
22. Passes through (-4, 5) and (1, -2)
4
23. x-intercept 7; y-intercept -5
−2
24. x-intercept -2; y-intercept 6
−4
25. Slope −
−6
26. Slope
y
10.
1
; x-intercept -6
3
27. Passes through (-3, 5); parallel to the line
y = −1
2
x
−2
3
; x-intercept 4
2
2
−2
4
28. Passes through (2, -6); parallel to the line
y=4
29. Passes through (5, -7); parallel to the line
y = −5 x + 3
−4
For each of the following equations,
MATH 1330 Precalculus
19
Exercise Set 2.1: Linear and Quadratic Functions
30. Passes through (5, -7); perpendicular to the line
y = −5 x + 3
31. Passes through (2, 3); parallel to the line
5x − 2 y = 6
32. Passes through (-1, 5); parallel to the line
4x + 3y = 8
33. Passes through (2, 3); perpendicular to the line
5x − 2 y = 6
34. Passes through (-1, 5); perpendicular to the line
4x + 3y = 8
35. Passes through (4, -6); parallel to the line
containing (3, -5) and (2, 1)
36. Passes through (8, 3) ; parallel to the line
containing (−2, − 3) and (−4, 6)
37. Perpendicular to the line containing (4, -2) and
(10, 4); passes through the midpoint of the line
segment connecting these points.
Answer the following, assuming that each situation
can be modeled by a linear function.
43. If a company can make 21 computers for
$23,000, and can make 40 computers for
$38,200, write an equation that represents the
cost of x computers.
44. A certain electrician charges a $40 traveling fee,
and then charges $55 per hour of labor. Write an
equation that represents the cost of a job that
takes x hours.
For each of the quadratic functions given below:
(a) Complete the square to write the equation in
the standard form f ( x ) = a( x − h)2 + k .
(b) State the coordinates of the vertex of the
parabola.
(c) Sketch the graph of the parabola.
(d) State the maximum or minimum value of the
function, and state whether it is a maximum
or a minimum.
(e) Find the axis of symmetry. (Be sure to write
your answer as an equation of a line.)
38. Perpendicular to the line containing (−3, 5) and
(7, − 1) ; passes through the midpoint of the line
segment connecting these points.
39.
f passes through ( −3, − 6 ) and f −1 passes
through ( −8, − 9 ) .
40.
f passes through ( 2, − 1) and f −1 passes
through ( 9, 4 ) .
41. The x-intercept for f is 3 and the x-intercept for
f −1 is −8 .
42. The y-intercept for f is 4 and the y-intercept
for f −1 is −6 .
45. f ( x) = x 2 + 6 x + 7
46. f ( x) = x 2 − 8 x + 21
47. f ( x) = x 2 − 2 x
48. f ( x) = x 2 + 10 x
49. f ( x) = 2 x 2 − 8 x + 11
50. f ( x) = 3 x 2 + 18 x + 15
51. f ( x) = − x 2 − 8 x − 9
52. f ( x) = − x 2 + 4 x − 7
53. f ( x) = −4 x 2 + 24 x − 27
54. f ( x) = −2 x 2 − 8 x − 14
55. f ( x) = x 2 − 5 x + 3
56. f ( x) = x 2 + 7 x − 1
57. f ( x) = 2 − 3 x − 4 x 2
58. f ( x) = 7 − x − 3x 2
20
University of Houston Department of Mathematics
Exercise Set 2.1: Linear and Quadratic Functions
Each of the quadratic functions below is written in the
form f ( x ) = ax 2 + bx + c . For each function:
68. f ( x) = −2 x 2 + 16 x + 40
69. f ( x) = −4 x 2 − 8 x + 5
(a) Find the vertex ( h, k ) of the parabola by using
the formulas h = − 2ba and k = f ( − 2ba ) .
(Note: When only the vertex is needed, this
method can be used instead of completing the
square.)
(b) State the maximum or minimum value of the
function, and state whether it is a maximum
or a minimum.
70. f ( x) = 4 x 2 − 16 x − 9
71. f ( x) = x 2 − 6 x + 3
72. f ( x) = x 2 + 10 x + 5
73. f ( x) = x 2 − 2 x + 5
74. f ( x) = x 2 + 4
59. f ( x) = x 2 − 12 x + 50
75. f ( x) = 9 − 4 x 2
60. f ( x) = − x 2 + 14 x − 10
76. f ( x) = 9 x 2 − 100
61. f ( x) = −2 x 2 + 16 x − 9
62. f ( x) = 3 x 2 − 12 x + 29
For each of the following problems, find a quadratic
function satisfying the given conditions.
63. f ( x) = −2 x 2 + 9 x + 3
77. Vertex (2, − 5) ; passes through (7, 70)
64. f ( x) = −6 x 2 + x − 5
78. Vertex (−1, − 8) ; passes through (2, 10)
The following method can be used to sketch a
reasonably accurate graph of a parabola without
plotting points. Each of the quadratic functions below
is written in the form f ( x ) = ax 2 + bx + c . The graph
of a quadratic function is a parabola with vertex,
where h = − 2ba and k = f
( − 2ba ) .
(a) Find all x-intercept(s) of the parabola by
setting f ( x ) = 0 and solving for x.
(b) Find the y-intercept of the parabola.
(c) Give the coordinates of the vertex (h, k) of the
parabola, using the formulas h = − 2ba and
k= f
( − 2ba ) .
(d) State the maximum or minimum value of the
function, and state whether it is a maximum
or a minimum.
(e) Find the axis of symmetry. (Be sure to write
your answer as an equation of a line.)
(f) Draw a graph of the parabola that includes
the features from parts (a) through (d).
2
65. f ( x) = x − 2 x − 15
79. Vertex (5, 7) ; passes through (3, 4)
80. Vertex (−4, 3) ; passes through (1, 13)
Answer the following.
81. Two numbers have a sum of 10. Find the largest
possible value of their product.
82. Jim is beginning to create a garden in his back
yard. He has 60 feet of fence to enclose the
rectangular garden, and he wants to maximize
the area of the garden. Find the dimensions Jim
should use for the length and width of the
garden. Then state the area of the garden.
83. A rocket is fired directly upwards with a velocity
of 80 ft/sec. The equation for its height, H, as a
function of time, t, is given by the function
H (t ) = −16t 2 + 80t .
(a) Find the time at which the rocket reaches its
maximum height.
(b) Find the maximum height of the rocket.
66. f ( x) = x 2 − 8 x + 16
67. f ( x) = 3 x 2 + 12 x − 36
MATH 1330 Precalculus
21
Exercise Set 2.1: Linear and Quadratic Functions
84. A manufacturer has determined that their daily
profit in dollars from selling x machines is given
by the function
P ( x) = −200 + 50 x − 0.1x 2 .
Using this model, what is the maximum daily
profit that the manufacturer can expect?
22
University of Houston Department of Mathematics
Exercise Set 2.2: Polynomial Functions
Answer True or False.
Answer the following.
(a) State whether or not each of the following
expressions is a polynomial. (Yes or No.)
(b) If the answer to part (a) is yes, then state the
degree of the polynomial.
(c) If the answer to part (a) is yes, then classify
the polynomial as a monomial, binomial,
trinomial, or none of these. (Polynomials of
four or more terms are not generally given
specific names.)
4 + 3x3
2.
6 x5 + 3x 3 +
8
x
3x − 5
4.
2 x3 + 4 x2 − 7 x − 4
5.
5x − 6x + 7
x2 − 4 x + 5
6.
8
7.
7
2
8.
7
5 3
+
− −2
x3 x 2 x
9.
3−1 x 4 − 7 −1 x + 2
24. (a) −3a 4 b5 is a fifth degree polynomial.
+ 2x
1
3
(b) −3a 4 b5 is a trinomial.
(c) −3a 4 b5 is a ninth degree polynomial.
(d) −3a 4 b5 is a monomial.
− 4x
1
Sketch a graph of each of the following functions.
2
25. P( x ) = x 3
2
11.
22. (a) x 2 − 4 x + 7 x 3 is a second degree
polynomial.
(b) x 2 − 4 x + 7 x 3 is a binomial.
(c) 3x 7 − 2 x 4 y 6 − 3 y8 is an eighth degree
polynomial.
(d) 3x 7 − 2 x 4 y 6 − 3 y8 is a trinomial.
x 2 − 53 x + 9
4
(d) 7 x − 2 x3 is a first degree polynomial
23. (a) 3x 7 − 2 x 4 y 6 − 3 y8 is a tenth degree
polynomial.
(b) 3x 7 − 2 x 4 y 6 − 3 y8 is a binomial.
2
1
(c) 7 x − 2 x3 is a binomial.
(d) x 2 − 4 x + 7 x 3 is a trinomial.
3.
10. −9 x
(b) 7 x − 2 x3 is a third degree polynomial.
(c) x 2 − 4 x + 7 x 3 is a third degree polynomial.
1.
3
21. (a) 7 x − 2 x3 is a trinomial.
x − 3x + 1
26. P( x ) = x 4
3
2
12. − x
13.
6
27. P( x ) = x 6
6 x3 + 8x 2
x
28. P( x ) = x 5
29. P( x ) = x n , where n is odd and n > 0.
14. −3 + 5 x 2 + 6 x 4 − 3x9
30. P( x ) = x n , where n is even and n > 0.
15. 3a 3b 4 − 2a 2 b2
16. −4 x5 y − 2 − 3x −4 y 9
Answer the following.
3
17. 4x y + 2
xy
5 3
18.
2
5
x 2 y 9 z + 3xy − 14 x3 y 4 z 2
19. −4xyz 3 − 52 y 7 − 73 x 4 y 3 z 2
31. The graph of P( x) = ( x − 1)( x − 2)3 ( x + 4) 2 has xintercepts at x = 1, x = 2, and x = −4.
(a) At and immediately surrounding the point
x = 2 , the graph resembles the graph of what
familiar function? (Choose one.)
y=x
7
3 5
6
y = x2
y = x3
2 4
20. − a + 2a b + b − 3a b
Continued on the next page…
MATH 1330 Precalculus
23
Exercise Set 2.2: Polynomial Functions
(b) At and immediately surrounding the point
x = −4 , the graph resembles the graph of
what familiar function? (Choose one.)
y=x
y = x2
y = x3
Choices for 33-38:
A.
B.
y
y
80
40
(c) If P(x ) were to be multiplied out
completely, the leading term of the
polynomial would be: (Choose one; do not
actually multiply out the polynomial.)
40
x
−4
−2
2
4
x
−4
−2
2
−40
x3 ; − x3 ; x 4 ; − x 4 ; x5 ; − x5 ; x 6 ; − x 6
32. The graph of Q ( x) = −( x + 3)2 ( x − 5)3 has xintercepts at x = −3 and x = 5.
(a) At and immediately surrounding the point
x = −3 , the graph resembles the graph of
what familiar function? (Choose one.)
y=x
y = x2
−40
C.
D.
400 y
y
10
x
−2
2
4
x
−400
−4
y = x3
−2
2
4
6
8
−800
(b) At and immediately surrounding the point
x = 5 , the graph resembles the graph of what
familiar function? (Choose one.)
y=x
y = x2
−10
−1200
y = x3
E.
(c) If P(x ) were to be multiplied out
completely, the leading term of the
polynomial would be: (Choose one; do not
actually multiply out the polynomial.)
F.
y
y
10
200
x
x
−6
−4
−2
2
4
6
x3 ; − x3 ; x 4 ; − x 4 ; x5 ; − x5 ; x 6 ; − x 6
−2
2
−200
−10
Match each of the polynomial functions below with its
graph. (The graphs are shown in the next column.)
33. P( x ) = ( x − 2)( x + 1)( x + 4)
For each of the functions below:
35. R( x) = −( x − 2)( x + 1) 2 ( x + 4) 2
(f) Find the x- and y-intercepts.
(g) Sketch the graph of the function. Be sure to
show all x- and y-intercepts, along with the
proper behavior at each x-intercept, as well as
the proper end behavior.
36. S ( x) = ( x − 2) 2 ( x + 1)( x + 4)
39. P( x) = ( x − 5)( x + 3)
34. Q ( x) = −( x + 2)( x − 1)( x − 4)
40. P( x ) = −( x − 3)( x + 1)
37. U ( x) = ( x + 2) 2 ( x − 1)3 ( x − 4)
38. V ( x) = −( x + 2)3 ( x − 1)3 ( x − 4) 2
41. P( x ) = −( x − 6) 2
42. P( x ) = ( x + 3) 2
43. P( x) = ( x − 5)( x + 2)( x + 6)
44. P( x) = 3 x( x − 4)( x − 7)
45. P( x ) = − 12 ( x − 4)( x − 1)( x + 3)
46. P( x ) = −( x + 6)( x − 2)( x − 5)
24
University of Houston Department of Mathematics
4
Exercise Set 2.2: Polynomial Functions
Polynomial functions can be classified according to
their degree, as shown below. (Linear and quadratic
functions have been covered in previous sections.)
47. P( x ) = ( x + 2) 2 ( x − 4)
48. P( x) = (5 − x)( x + 3) 2
49. P( x ) = (3 x − 2)( x + 4)( x − 5)( x + 1)
50.
Degree
0 or 1
2
3
4
5
n
P ( x ) = − 13 ( x + 5)( x + 1)( x + 3)( x − 2)
51. P( x ) = x( x + 2)(4 − x)( x + 6)
52. P( x) = ( x − 1)( x − 3)( x + 2)( x + 5)
53. P( x) = ( x − 3)2 ( x + 4) 2
54. P( x ) = − x(2 x − 5)3
Answer the following.
55. P( x ) = ( x + 5)3 ( x − 4)
75. Write the equation of the cubic polynomial P( x )
that satisfies the following conditions:
56. P( x ) = x 2 ( x − 6) 2
57. P( x ) = ( x + 3) 2 ( x − 4)3
P ( −4 ) = P (1) = P ( 3) = 0 , and P (0) = −6 .
58. P( x ) = −2 x(3 − x )3 ( x + 1)
2
2
59. P( x ) = − x( x − 2) ( x + 3) ( x − 4)
60. P( x ) = ( x − 5)3 ( x − 2) 2 ( x + 1)
61. P( x) = x8 ( x − 1)6 ( x + 1)7
3
Name
Linear
Quadratic
Cubic
Quartic
Quintic
nth degree
polynomial
4
62. P( x ) = − x ( x + 1) ( x − 1)
7
76. Write an equation for a cubic polynomial P( x )
with leading coefficient −1 whose graph passes
through the point ( 2, 8 ) and is tangent to the xaxis at the origin.
77. Write the equation of the quartic polynomial
with y-intercept 12 whose graph is tangent to the
x-axis at ( −2, 0 ) and (1, 0 ) .
63. P( x ) = x3 − 6 x 2 + 8 x
64. P( x ) = x3 − 2 x 2 − 15 x
65. P( x ) = 25 x − x 3
66. P( x ) = −3x 3 − 5 x 2 + 2 x
67. P( x) = − x 4 + x3 + 12 x 2
68. P( x) = x 4 − 16 x 2
78. Write the equation of the sixth degree
polynomial with y-intercept −3 whose graph is
tangent to the x-axis at ( −2, 0 ) , ( −1, 0 ) , and
( 3, 0 ) .
Use transformations (the concepts of shifting,
reflecting, stretching, and shrinking) to sketch each of
the following graphs.
69. P( x ) = x5 − 9 x3
79. P( x ) = x3 + 5
70. P( x ) = − x5 − 3 x 4 + 18 x3
80. P( x ) = − x3 − 2
71. P( x) = x3 + 4 x 2 − x − 4
81. P( x ) = −( x − 2)3 + 4
72. P( x ) = x3 − 5 x 2 − 4 x + 20
82. P( x) = ( x + 5)3 − 1
73. P( x ) = x 4 − 13 x 2 + 36
83. P( x ) = 2 x 4 − 3
74. P( x ) = x 4 − 17 x 2 + 16
84. P( x ) = −( x − 2) 4 + 5
85. P( x ) = −( x + 1)5 − 4
86. P( x) = ( x + 3)5 + 2
MATH 1330 Precalculus
25
Exercise Set 2.3: Rational Functions
Recall from Section 1.2 that an even function is
symmetric with respect to the y-axis, and an odd function
is symmetric with respect to the origin. This can
sometimes save time in graphing rational functions. If a
function is even or odd, then half of the function can be
graphed, and the rest can be graphed using symmetry.
8.
y
6
4
2
x
−6
Determine if the functions below are even, odd, or
neither.
1.
f ( x) =
−2
2
4
6
−2
5
x
−4
3
x −1
2.
f ( x) = −
3.
f ( x) =
4
x2 − 9
4.
f ( x) =
9x2 − 1
x4
5.
f ( x) =
x −1
x2 − 4
f ( x) =
7
x3
6.
−4
−6
(Notice the asymptotes at x = 0 and y = 0 .)
For each of the following graphs:
In each of the graphs below, only half of the graph is
given. Sketch the remainder of the graph, given that the
function is:
(a) Even
(b) Odd
(h) Identify the location of any hole(s)
(i.e. removable discontinuities)
(i) Identify any x-intercept(s)
(j) Identify any y-intercept(s)
(k) Identify any vertical asymptote(s)
(l) Identify any horizontal asymptote(s)
9.
10 y
8
6
4
7.
6
y
2
x
−10 −8 −6 −4 −2
−2
4
2
4
6
8
−4
−6
2
−8
x
−6
−4
−2
2
4
6
−2
y
10.
6
−4
4
2
x
−6
−8 −6 −4 −2
2
4
6
8
−2
(Notice the asymptotes at x = 2 and y = 0 .)
−4
−6
−8
26
University of Houston Department of Mathematics
Exercise Set 2.3: Rational Functions
For each of the following rational functions:
(a) Find the domain of the function
(b) Identify the location of any hole(s)
(i.e. removable discontinuities)
(c) Identify any x-intercept(s)
(d) Identify any y-intercept(s)
(e) Identify any vertical asymptote(s)
(f) Identify any horizontal asymptote(s)
(g) Identify any slant asymptote(s)
(h) Sketch the graph of the function. Be sure to
include all of the above features on your graph.
11. f ( x) =
12. f ( x) =
3
x −5
−4
x+7
24. f ( x) =
x3
2 x 2 − 18
25. f ( x) =
−(3x + 5)( x − 2)
x ( x − 2)
26. f ( x) =
−( x + 4)(5 x − 7)
( x − 3)( x + 4)
27. f ( x) =
28. f ( x) =
29.
2 x 2 − 18
x2 + 4x + 3
8 x 2 − 16 x
5 x 2 − 20
16 − x 4
2 x3
x3 − 2 x 2 − x + 2
x2 − 4
2x + 3
f ( x) =
x
30. f ( x) =
14. f ( x) =
9 − 4x
x
31. f ( x) =
15. f ( x) =
x−6
x+3
32. f ( x) =
16. f ( x) =
x+5
x−2
33. f ( x) =
17. f ( x) =
−4 x + 8
2x + 3
34. f ( x) =
18. f ( x ) =
3x + 6
2x −1
35. f ( x) =
19. f ( x) =
( x − 2)( x + 3)
( x − 2)( x − 4)
36. f ( x) =
20. f ( x) =
( x + 3)(6 − x)
( x − 2)( x + 3)
37. f ( x) =
21. f ( x) =
x 2 + x − 20
x−4
38. f ( x) = −
22. f ( x) =
x 2 − 3 x − 10
x−5
39. f ( x) =
x3 + 2 x 2 − 9 x − 18
x2
23. f ( x) =
4 x3
x2 − 1
40. f ( x) =
x 4 − 10 x 2 + 9
x3
13.
MATH 1330 Precalculus
8
x2 − 4
−12
x2 + x − 6
6x − 6
2
x − x − 12
−8 x − 16
2
x + 2 x − 15
( x − 3)( x + 2)( x − 4)
( x + 1)( x − 4)( x − 2)
2 x3 + 10 x 2
3
x + 5 x 2 − 9 x − 45
x( x − 5)( x + 1)( x − 3)
( x + 1)( x − 3)
( x − 4)( x − 3)( x − 2)( x + 1)
( x − 4)( x − 2)
27
Exercise Set 2.3: Rational Functions
Answer the following.
41. In the function f ( x ) =
x2 − 5x + 3
3x2 + 2 x − 3
(a) Use the quadratic formula to find the xintercepts of the function, and then use a
calculator to round these answers to the nearest
tenth.
(b) Use the quadratic formula to find the vertical
asymptotes of the function, and then use a
calculator to round these answers to the nearest
tenth.
42. In the function f ( x ) =
2 x2 + 7 x − 1
x2 − 6 x + 4
(a) Use the quadratic formula to find the xintercepts of the function, and then use a
calculator to round these answers to the nearest
tenth.
(b) Use the quadratic formula to find the vertical
asymptotes of the function, and then use a
calculator to round these answers to the nearest
tenth.
47. f ( x) =
4 x 2 + 12 x + 9
x2 − x + 7
48. f ( x) =
x2 + 5x − 1
5 x 2 − 10 x − 3
Answer the following.
49. The function f ( x) =
6x − 6
2
x − x − 12
was graphed in
Exercise 33.
(a) Find the point of intersection of f ( x ) and the
horizontal asymptote.
(b) Sketch the graph of f ( x ) as directed in
Exercise 33, but also label the intersection of
f ( x ) and the horizontal asymptote.
50. The function f ( x) =
−8 x − 16
x 2 + 2 x − 15
was graphed in
Exercise 34.
(a) Find the point of intersection of f ( x ) and the
horizontal asymptote.
The graph of a rational function never intersects a
vertical asymptote, but at times the graph intersects a
horizontal asymptote. For each function f ( x ) below,
(a) Find the equation for the horizontal asymptote of
the function.
(b) Sketch the graph of f ( x ) as directed in
Exercise 34, but also label the intersection of
f ( x ) and the horizontal asymptote.
(b) Find the x-value where f ( x ) intersects the
horizontal asymptote.
(c) Find the point of intersection of f ( x ) and the
horizontal asymptote.
28
43. f ( x ) =
x2 + 2 x + 3
x2 − x − 3
44. f ( x) =
x2 + 4 x − 2
x2 − x − 7
45. f ( x) =
x2 + 2 x + 3
2x2 + 6x − 1
46. f ( x ) =
3x 2 + 5 x − 1
x2 − x − 3
University of Houston Department of Mathematics
Exercise Set 2.4: Applications and Writing Functions
Answer the following. If an example contains units of
measurement, assume that any resulting function
reflects those units. (Note: Refer to Appendix A.3 if
necessary for a list of Geometric Formulas.)
1.
The perimeter of a rectangle is 54 feet.
(a) Express its area, A, as a function of its
width, w.
(b) For what value of w is A the greatest?
(c) What is the maximum area of the rectangle?
2.
The perimeter of a rectangle is 62 feet.
(a) Express its area, A, as a function of its
length, .
(b) For what value of is A the greatest?
(c) What is the maximum area of the rectangle?
3.
Two cars leave an intersection at the same time.
One is headed south at a constant speed of 50
miles per hour. The other is headed east at a
constant speed of 120 miles per hour. Express
the distance, d, between the two cars as a
function of the time, t.
4.
5.
Two cars leave an intersection at the same time.
One is headed north at a constant speed of 30
miles per hour. The other is headed west at a
constant speed of 40 miles per hour. Express the
distance, d, between the two cars as a function of
the time, t.
If the sum of two numbers is 20, find the
smallest possible value of the sum of their
squares.
6.
If the sum of two numbers is 16, find the
smallest possible value of the sum of their
squares.
7.
If the sum of two numbers is 8, find the largest
possible value of their product.
8.
If the sum of two numbers is 14, find the largest
possible value of their product.
9.
A farmer has 1500 feet of fencing. He wants to
fence off a rectangular field that borders a
straight river (needing no fence along the river).
What are the dimensions of the field that has the
largest area?
MATH 1330 Precalculus
10. A farmer has 2400 feet of fencing. He wants to
fence off a rectangular field that borders a
straight river (needing no fence along the river).
What are the dimensions of the field that has the
largest area?
11. A farmer with 800 feet of fencing wants to
enclose a rectangular area and divide it into 3
pens with fencing parallel to one side of the
rectangle. What is the largest possible total area
of the 3 pens?
12. A farmer with 1800 feet of fencing wants to
enclose a rectangular area and divide it into 5
pens with fencing parallel to one side of the
rectangle. What is the largest possible total area
of the 5 pens?
13. The hypotenuse of a right triangle is 6 m.
Express the area, A, of the triangle as a function
of the length x of one of the legs.
14. The hypotenuse of a right triangle is 11 m.
Express the area, A, of the triangle as a function
of the length x of one of the legs.
15. The area of a rectangle is 22 ft2. Express its
perimeter, P, as a function of its length, .
16. The area of a rectangle is 36 ft2. Express its
perimeter, P, as a function of its width, w.
17. A rectangle has a base on the x-axis and its upper
two vertices on the parabola y = 9 − x 2 .
(a) Express the area, A, of the rectangle as a
function of x.
(b) Express the perimeter, P, of the rectangle as
a function of x.
18. A rectangle has a base on the x-axis and its lower
two vertices on the parabola y = x 2 − 16 .
(a) Express the area, A, of the rectangle as a
function of x.
(b) Express the perimeter, P, of the rectangle as
a function of x.
19. In a right circular cylinder of radius r, if the
height is twice the radius, express the volume, V,
as a function of r.
29
Exercise Set 2.4: Applications and Writing Functions
20. In a right circular cylinder of radius r, if the
height is half the radius, express the volume, V,
as a function of r.
21. A right circular cylinder of radius r has a volume
of 300 cm3.
(a) Express the lateral area, L, in terms of r.
(b) Express the total surface area, S, as a
function of r.
22. A right circular cylinder of radius r has a volume
of 750 cm3.
(a) Express the lateral area, L, in terms of r.
(b) Express the total surface area, S, as a
function of r.
23. In a right circular cone of radius r, if the height is
four times the radius, express the volume, V, as a
function of r.
24. In a right circular cone of radius r, if the height is
five times the radius, express the volume, V, as a
function of r.
25. An open-top box with a square base has a
volume of 20 cm3. Express the surface area, S, of
the box as a function of x, where x is the length
of a side of the square base.
(b) What are the dimensions of the box with the
largest surface area?
(c) What is the maximum surface area of the
box?
29. A wire of length x is bent into the shape of a
circle.
(a) Express the circumference, C, in terms of x.
(b) Express the area of the circle, A, as a
function of x.
30. A wire of length x is bent into the shape of a
square.
(a) Express the area, A, of the square as a
function of x.
(b) Express the diagonal, d, of the square as a
function of x.
31. Let P ( x, y ) be a point on the graph of
y = x 2 − 10 .
(a) Express the distance, d, from P to the origin
as a function of x.
(b) Express the distance, d, from P to the point
( 0, − 2 ) as a function of x.
32. Let P ( x, y ) be a point on the graph of
y = x2 − 7 .
26. An open-top box with a square base has a
volume of 12 cm3. Express the surface area, S, of
the box as a function of x, where x is the length
of a side of the square base.
27. A piece of wire 120 cm long is cut into several
pieces and used to construct the skeleton of a
rectangular box with a square base.
(a) Express the surface area, S, of the box in
terms of x, where x is the length of a side of
the square base.
(b) What are the dimensions of the box with the
largest surface area?
(c) What is the maximum surface area of the
box?
28. A piece of wire 96 in long is cut into several
pieces and used to construct the skeleton of a
rectangular box with a square base.
(a) Express the surface area, S, of the box in
terms of x, where x is the length of a side of
the square base.
30
(a) Express the distance, d, from P to the origin
as a function of x.
(b) Express the distance, d, from P to the point
( 0, 5 ) as a function of x.
33. A circle of radius r is inscribed in a square.
Express the area, A, of the square as a function of
r.
34. A square is inscribed in a circle of radius r.
Express the area, A, of the square as a function of
r.
35. A rectangle is inscribed in a circle of radius 4
cm.
(a) Express the perimeter, P, of the rectangle in
terms of its width, w.
(b) Express the area, A, of the rectangle in terms
of its width, w.
University of Houston Department of Mathematics
Exercise Set 2.4: Applications and Writing Functions
36. A rectangle is inscribed in a circle of diameter 20
cm.
(a) Express the perimeter, P, of the rectangle in
terms of its length, .
(b) Express the area, A, of the rectangle in terms
of its length, .
43. An equilateral triangle is inscribed in a circle of
radius r, as shown below. Express the
circumference, C, of the circle as a function of
the length, x, of a side of the triangle.
x
r
37. An isosceles triangle has fixed perimeter P (so P
is a constant).
(a) If x is the length of one of the two equal
sides, express the area, A, as a function of x.
(b) What is the domain of A ?
38. Express the volume, V, of a sphere of radius r as
a function of its surface area, S.
39. Two cars are approaching an intersection. One is
2 miles south of the intersection and is moving at
a constant speed of 30 miles per hour. At the
same time, the other car is 3 miles east of the
intersection and is moving at a constant speed of
40 miles per hour.
(a) Express the distance, d, between the cars as
a function of the time, t.
(b) At what time t is the value of d the smallest?
44. An equilateral triangle is inscribed in a circle of
radius r, as shown below. Express the area, A,
within the circle, but outside the triangle, as a
function of the length, x, of a side of the triangle.
x
r
40. Two cars are approaching an intersection. One is
5 miles north of the intersection and is moving at
a constant speed of 40 miles per hour. At the
same time, the other car is 6 miles west of the
intersection and is moving at a constant speed of
30 miles per hour.
(a) Express the distance, d, between the cars as
a function of the time, t.
(b) At what time t is the value of d the smallest?
41. A straight wire 40 cm long is bent into an L
shape. What is the shortest possible distance
between the two ends of the bent wire?
42. A straight wire 24 cm long is bent into an L
shape. What is the shortest possible distance
between the two ends of the bent wire?
MATH 1330 Precalculus
31
Exercise Set 3.1: Exponential Functions
Solve for x. (If no solution exists, state ‘No solution.’)
1.
(a) 3x+ 2 = 0
(b) 32 x− 7 = 1
2.
(a) 7 x+ 3 = 1
(b) 7 x = 0
3.
(a) 53 x−1 = 125
(b) 253 x−1 = 125
4.
(a) 25 x− 2 = 32
(b) 85 x− 2 = 32
5.
(a) 24 x−1 − 8 = 0
(b) 2 x−3 + 8 = 0
6.
(a) 7 x+ 3 −
1
=0
7
(b) 7 2 x + 3 = 52
14. Based on your answers to 9-12 above:
(a) Draw a sketch of f ( x) = 2.7 x without
plotting points.
(b) Draw a sketch of f ( x) = 0.73x without
plotting points.
(c) What point on the graph do parts (a) and (b)
have in common?
(d) Name any asymptote(s) for parts (a) and (b).
Answer the following.Graph each of the following
pairs of functions on the same set of axes.
15. (a) Graph the functions f ( x) = 6 x and
x
7.
8.
(a) 27 x+ 5 = 3
(b) 36 3 x+ 0.5 =
(a) 252 x−1 = 3 5
(b)
1
6
7
= 7 2 x−1
49
1
g ( x ) = on the same set of axes.
6
(b) Compare the graphs drawn in part (a). What
is the relationship between the graphs?
(c) Explain the result from part (b)
algebraically.
x
Graph each of the following functions by plotting
points. Show any asymptote(s) clearly on your graph.
5
16. (a) Graph the functions f ( x) = and
2
x
9.
2
g ( x ) = on the same set of axes.
5
(b) Compare the graphs drawn in part (a). What
is the relationship between the graphs?
(c) Explain the result from part (b)
algebraically.
f ( x) = 3x
10. f ( x) = 5 x
1
11. f ( x) =
2
x
4
12. f ( x) =
3
x
Sketch a graph of each of the following functions.
(Note: You do not need to plot points.) Be sure to label
at least one key point on your graph, and show any
asymptotes.
17. f ( x) = 8 x
Answer the following.
13. Based on your answers to 9-12 above:
(a) Draw a sketch of f ( x) = a x , where a > 1 .
18. f ( x) = 12 x
19. f ( x) = 0.2 x
x
(b) Draw a sketch of f ( x) = a , where
0 < a < 1.
(c) What point on the graph do parts (a) and (b)
have in common?
(d) Name any asymptote(s) for parts (a) and (b).
32
20. f ( x) = 1.4 x
7
21. f ( x) =
2
x
3
22. f ( x) =
8
x
University of Houston Department of Mathematics
Exercise Set 3.1: Exponential Functions
For each of the following examples,
(a) Find any intercept(s) of the function.
(b) Use transformations (the concepts of
reflecting, shifting, stretching, and shrinking)
to sketch the graph of the function. Be sure to
label the transformation of the point ( 0, 1) .
Clearly label any intercepts and/or
asymptotes.
(c) State the domain and range of the function.
(d) State whether the graph is increasing or
decreasing.
39. f ( x ) = 8−1− x − 32
40. f ( x ) = 51− x − 1
41. f ( x) = −6− x − 36
42. f ( x) = −2− x + 16
43. f ( x) = 2 ⋅ 3x
44. f ( x) = 12 ⋅ 5 x
23. f ( x) = 2 x −1
24. f ( x) = 3x + 2
Find the exponential function of the form
f ( x ) = C ⋅ a x which satisfies the given conditions.
25. f ( x) = −5x
45. Passes through the points ( 0, 4 ) and ( 3, 32 ) .
3
26. f ( x) = −
4
x
46. Passes through the points ( 0, 3) and ( 2, 108) .
(
)
27. f ( x) = 8− x
47. Passes through the points ( 0, 2 ) and − 1, 72 .
28. f ( x) = 10− x
7 .
48. Passes through the points ( 0, 7 ) and − 2, 16
(
)
x
1
29. f ( x) = − 4
2
30. f ( x) = 4 x − 8
True or False? (Answer these without using a
calculator.)
49. e 2 < 9
31. f ( x) = 9 x + 2 − 27
50. e 2 > 4
32. f ( x) = 7 x −3 + 7
51.
e<2
33. f ( x ) = −4 x +1 + 2
52.
e >1
34. f ( x ) = −8 x − 2 + 32
53. e −1 < 0
35. f ( x) = 9 − 3x + 4
36. f ( x) = −25 − 5
x+3
54. e −1 > 0
55. e3 > 27
56. e <
2− x
37. f ( x) = 3
38. f ( x) = 4
−3− x
5
2
57. e0 = 1
58. e1 = 0
MATH 1330 Precalculus
33
Exercise Set 3.1: Exponential Functions
Answer the following.
59. Graph f ( x) = 2 x and g ( x ) = e x on the same set
of axes.
71. f ( x) = e − x − 4 − 3
72. f ( x) = e − x +3 + 2
60. Graph f ( x) = e x and g ( x ) = 3x on the same set
of axes.
61. Sketch the graph of f ( x) = e x and then find the
following:
(a)
(b)
(c)
(d)
Domain
Range
Horizontal Asymptote
y-intercept
62. Sketch the graph of f ( x) = 3x and then find the
following:
(a)
(b)
(c)
(d)
(e)
Domain
Range
Horizontal Asymptote
y-intercept
How do your answers compare with those in
the previous question?
Graph the following functions, not by plotting points,
but by applying transformations to the graph of
y = e x . Be sure to label at least one key point on your
graph (the transformation of the point (0, 1)) and show
any asymptotes. Then state the domain and range of
the function.
63. f ( x) = −3e x
64. f ( x) = 2e− x
65. f ( x) = e x −1 − 6
66. f ( x) = e x + 2 + 1
67. f ( x) = e− x + 3
68. f ( x) = −e x − 2
69. f ( x) = −4e x − 5
70. f ( x) = 2e x + 6 − 4
34
University of Houston Department of Mathematics
Exercise Set 3.2: Logarithmic Functions
Write each of the following equations in exponential
form.
( )
(b) ln e−2
( )
(b) ln
26. (a) ln
( e)
4
(b) ln (1)
27. (a) ln
( )
1
e4
(b) ln e x
28. (a) ln
( )
(b) ln e2 x
(a) log3 ( 81) = 4
(b) log 4 ( 8) =
2.
(a) log5 (1) = 0
(b) log 3 ( 19 ) = −2
25. (a) ln e−6
3.
(a) log 5 ( 7 ) = x
(b) ln ( x ) = 8
4.
(a) log x ( 2 ) = 3
(b) ln ( x ) = 2
Write each of the following equations in logarithmic
form.
5.
1
(b) 10−4 = 10,000
(a) 60 = 1
−3
6.
(a) 53 = 125
(b) 25
7.
(a) 7 1 = 7
(b) e 4 = x
8.
(a) 100.5 = 10
(b) e −5 = y
9.
x
(a) e = 7
(b) e
10. (a) e x = 12
2
x−2
1
= 125
=y
( )
24. (a) ln e5
1.
3
2
( )
(b) ln e4
23. (a) ln ( e )
( e)
3
( )
( )
1
e
Simplify each of the following expressions.
29. (a) 7
log
7
(5)
(b) 10
log
( 3)
log 5 ( 0.71)
log 4
30. (a) 10 ( )
(b) 5
ln 6
31. (a) e ( )
(b) e
ln 2
32. (a) e ( )
ln x
(b) e ( )
( )
ln x 4
(b) e5 x = y − 6
Simplify each of the following expressions.
Find the value of x in each of the following equations.
Write all answers in simplest form. (Some answers
may contain e.)
11. (a) log 2 ( 8 )
(b) log5 (1)
33. (a) log 2 ( 32 ) = x
(b) log x ( 9 ) = 2
12. (a) log3 ( 81)
(b) log8 ( 8 )
34. (a) log 5 (125 ) = x
(b) log x ( 64 ) = 3
35. (a) log 7 ( x ) = 2
(b) log x ( 3) =
36. (a) log3 ( x ) = 4
(b) log x ( 18 ) = −1
37. (a) log 25 (125 ) = x
(b) log16 ( x ) =
38. (a) log16 ( 32 ) = x
(b) log36 ( x ) = 0.5
39. (a) log x − 3 = 2
(b) log3 ( log 4 ( x ) ) = 0
( )
2
13. (a) log 7 ( 7 )
(b) log 6 6
14. (a) log9 (1)
(b) log 2 25
15. (a) log16 ( 4 )
(b) log 7 ( 17 )
16. (a) log (
1
10
)
( )
(b) log 32 ( 2 )
17. (a) log 27 ( 3)
1
(b) log 9 ( 81
)
18. (a) log 49 ( 7 )
(b) log 2 ( 18 )
19. (a) log 4 ( 0.25 )
(b) log
( 10 )
4
( 7)
(b) log5 ( 0.2 )
21. (a) log 4 ( 0.5 )
(b) log8 ( 32 )
20. (a) log 7
22. (a) log 25
6
( 5)
MATH 1330 Precalculus
(b)
(
)
1
2
1
4
40. (a) log x 2 − 9 = 3
(b) log5 ( log 2 ( x ) ) = 1
41. (a) ln ( x ) = 4
(b) ln ( x + 5 ) = 2
42. (a) ln ( x ) = 0
(b) ln x − 2 = 5
( )
log 32 161
35
Exercise Set 3.2: Logarithmic Functions
Solve for x. Give an exact answer, and then use a
calculator to round that answer to the nearest
thousandth. (Hint: Write the equations in
logarithmic form first.)
For each of the following examples,
(a) Use transformations (the concepts of
reflecting, shifting, stretching, and shrinking)
to sketch the graph of y = log b ( x ) . Be sure to
43. (a) 10 x = 20
(b) e x = 20
label the transformation of the point ( 1, 0 ) .
44. (a) 10 x = 45
(b) e x = 45
45. (a) 10 x =
2
5
(b) e x =
2
5
Clearly label any asymptotes.
(b) State the domain and range of the function.
(c) State whether the graph is increasing or
decreasing.
46. (a) 10 x =
4
7
(b) e x =
4
7
55. f ( x) = log 2 ( x ) + 3
47. 103 x− 2 = 6
56. f ( x) = log 5 ( x ) − 4
48. e5 x +1 = 7
57. f ( x) = − log3 ( x )
49. e
4 x2
50. 10 x
= 40
2 −1
=7
58. f ( x) = log 6 (− x)
59. f ( x) = log 7 ( x − 5)
60. f ( x) = log8 ( x + 2)
Answer the following.
51. (a) Sketch the graph of f ( x) = 2 x .
(b) Sketch the graph of f −1 ( x) on the same set
of axes.
(c) The inverse of y = 2 x is y = ________ .
52. (a) Sketch the graph of f ( x) = e x .
(b) Sketch the graph of f −1 ( x) on the same set
of axes.
(c) The inverse of y = e x is y = ________ .
61. f ( x) = log5 ( x + 1) − 4
62. f ( x) = log 6 ( x − 4) + 2
63. f ( x) = ln(− x)
64. f ( x) = − ln ( x )
65. f ( x) = − ln( x + 4)
66. f ( x) = ln(− x) − 1
67. f ( x) = 5 − ln(− x)
68. f ( x) = −3 − ln( x + 2)
53. (a) Sketch the graph of y = log b ( x ) , where
b > 1.
(b) Name any intercept(s) for the graph of
y = log b ( x ) .
(c) Name any asymptote(s) for the graph of
y = log b ( x ) .
Find the domain of each of the following functions.
(Note: You should be able to do this algebraically,
without sketching the graph.)
69. f ( x) = log8 ( x )
70. f ( x) = − log5 ( x )
54. (a) Sketch the graph of y = ln ( x ) .
(b) y = ln ( x ) can be written as y = log b ( x ) ,
where b = ___ .
(c) Name any intercept(s) for the graph of
y = ln ( x ) .
(d) Name any asymptote(s) for the graph of
y = ln ( x ) .
36
71. f ( x) = 2 ln ( x ) + 3
72. f ( x) = −3ln(2 x)
73. f ( x) = − log 6 ( x + 2)
74. f ( x) = log10 ( x − 3)
75. f ( x) = log10 (− x) + 3
University of Houston Department of Mathematics
Exercise Set 3.2: Logarithmic Functions
76. f ( x) = log3 (3 − 4 x ) + 5
77. f ( x) = ln(5 − 2 x) − 1
78. f ( x) = ln(−4 x) − 8
(
)
(
)
79. f ( x) = log 4 x 2 + 1
80. f ( x) = log 7 x 2 − 9
(
81. f ( x) = log 2 x 2 − x − 6
(
82. f ( x) = log 9 x 2 + 4
)
)
MATH 1330 Precalculus
37
Exercise Set 3.3: Laws of Logarithms
True or False?
(Note: Assume that x > 0, y > 0 , and x > y , so that
each logarithm below is defined.)
1.
log( xy ) = log ( x ) + log ( y )
2.
log ( x − y ) = log ( x ) − log ( y )
3.
log ( x + y ) = log ( x ) + log ( y )
4.
x
log = log ( x ) − log ( y )
y
5.
log ( x ) − log ( y ) =
6.
log ( x ) + log ( y ) = ( log x )( log y )
log ( x )
log ( y )
Rewrite each of the following expressions so that your
answer contains sums, differences, and/or multiples of
logarithms. Your answer should not contain logarithms of
any product, quotient, or power.
21. log 5 (9CD )
22. log3 ( FGH )
KP
23. log 7
L
7 RT
24. log8
M
(
)
(
)
25. ln B 2 P3 6 K
3
7.
ln ( x ) = 3ln ( x )
8.
ln x5 = 5 ln ( x )
9.
( ) = 4 ln ( x )
26. ln Y 5 X 4 Z
( )
ln x
4
7
10. ln ( x ) = 7 ln ( x )
x log 7 ( x )
11. log 7 =
y log 7 ( y )
12. log5 ( xy ) = log5 ( x ) ⋅ log 5 ( y )
13. log8 ( 0 ) = 1
14. log8 (1) = 0
9 k 2 m3
27. log 4
p w
7r 5 p 4
28. log
3
xyz
29. ln x ( y + 3)
30. ln 4 C ( J − M )
31. log3
(
32. log 9
5
4
7x
)
15. ln (1) = 0
16. ln ( 0 ) = 1
17. log 7 ( 2 ) =
18. log 7 ( 2 ) =
19. log 5 ( 4 ) =
20. log 5 ( 4 ) =
38
ln ( 2 )
ln ( 7 )
ln ( 7 )
ln ( 2 )
log ( 5 )
log ( 4 )
log ( 4 )
log ( 5 )
P
Q
x4 ( x − 5)
33. ln
x + 3
x 7 x3 + 2
34. ln 3
x−6
(
35. log
5
)
x2 − 7
(x
2
)
+ 3 ( x − 4)
2
University of Houston Department of Mathematics
Exercise Set 3.3: Laws of Logarithms
36. log
( x + 2 )3
( x3 − 5) ( x + 7 )4
46. 4 log 2 ( 3) − 2 log 2 ( 5 ) + 1
x2 − 9
x2 + 6x + 8
47. ln
+ ln
x−3
x+4
x 2 − 5 x − 35
x2 − 4 x
48. ln
− ln
x−7
x
Answer the following.
37. (a) Rewrite the following expression as a single
logarithm: log5 ( A ) + log 5 ( B ) − log5 ( C ) .
(b) Rewrite the number 1 as a logarithm with base
5.
(c) Use the result from part (b) to rewrite the
following expression as a single logarithm:
log5 ( A ) + log 5 ( B ) − log5 ( C ) + 1
(d) Use the result from part (b) to rewrite the
following expression as a single logarithm:
log5 ( A ) + log 5 ( B ) − log5 ( C ) − 1
38. (a) Rewrite the following expression as a single
logarithm: ln ( x ) − ln ( y ) − ln ( z ) .
(b) Rewrite the number 1 as a natural logarithm.
(c) Use the result from part (b) to rewrite the
following expression as a single logarithm:
ln ( x ) − ln ( y ) − ln ( z ) + 1
(d) Use the result from part (b) to rewrite the
following expression as a single logarithm:
ln ( x ) − ln ( y ) − ln ( z ) − 1
Rewrite each of the following expressions as a single
logarithm.
(
1
6
5 ln x + 2 ln ( x + 3) − 4 ln x 2 + 5 − 1
50.
1
8
2 ln ( x − 3) + 7 ln ( x ) − 12 9 ln ( x − 2 ) + ln x 2 − 5
(
51. log 2 ( 24 ) − log 2 ( 3)
52. log12 ( 3) + log12 ( 48)
53. log3 ( 45 ) − log3 ( 5 ) + log3
54. log 6 ( 9 ) + log 6 ( 4 ) + log 6
58. log ( 3, 000 ) − log 3
41. 4 log ( K ) + 3log ( P ) − 12 log ( Q )
59. log3
(
60. log 4
( 16 )
)
43. 3log 7 ( x − 2 ) − 15 log 7 x 2 − 3 − 4 log 7 ( x + 5) + 1
)
( 6)
( )
42. 7 log ( F ) − 14 log (V ) + log ( R )
(
4
log 2 (160 ) − log 2 (10 )
56.
log 2 ( 8 )
40. ln ( B ) − ln ( C ) − ln ( D ) − 1
1
2
( 3)
55. log 4 (10 ) + log 4 ( 6.4 )
39. ln (Y ) − ln ( Z ) + ln (W )
44.
)
Evaluate the following using the laws of logarithms.
Simplify your answer as much as possible. (Note: These
should be done without a calculator.)
log ( 5 ) + log ( 2 )
57.
log 3 10
(
)
49.
log 6 x 4 + 2 + 8 log 6 ( 7 − x ) − 2 log 6 ( x )
27
)
3
log 5 5 + log5 1
61.
3
log 2 2
( )
( )
45. log 7 ( 5 ) + 3log 7 ( 2 )
MATH 1330 Precalculus
39
Exercise Set 3.3: Laws of Logarithms
62. log5
( 5 ) + log
5
(1) − log 2 ( 3 2 )
write your answer as a decimal, correct to the nearest
thousandth.
2ln 5
63. e ( )
79. (a) log 5 ( 2 )
(b) log 6 (17.2 )
3ln 2
64. e ( )
80. (a) log 3 ( 7 )
(b) log8 ( 23.5 )
65. 10
66. 10
4 log ( 3)
Use the Change of Base Formula to write the following in
terms of common (base 10) logarithms. Then use a
calculator to write your answer as a decimal, correct to
the nearest thousandth.
2 log ( 7 )
( )
67. log3 917
(
90
68. log5 125
69.
)
81. (a) log 9 ( 4 )
(b) log 4 ( 8.9 )
82. (a) log 7 ( 8 )
(b) log 2 ( 0.7 )
3ln 4
e ( )
ln
( e)
5
103log ( 2)
70.
log 2 3 16
( )
71. 7
log 7 ( 6 ) + log 7 ( 5 )
Evaluate the following. (Note: These should be done
without a calculator.)
83. Find the value of the following:
log 2 ( 3) log 3 ( 5) log5 ( 8 )
84. Find the value of the following:
log 27 ( 36 ) log 36 ( 49 ) log 49 ( 81)
log 3 ( 5) − log3 ( 7 )
72. 3
85. Find the value of the following:
log3 ( 3) log3 ( 9 ) log 3 ( 27 ) log 3 ( 81)
Use the laws of logarithms to express y as a function of x.
86. Find the value of the following:
73. ln ( y ) = 3ln ( x )
1
74. log ( y ) − log ( x ) = 0
3
log5 ( 5 ) log 5 ( 25 ) log 5 (125 )
Rewrite the following as sums so that each logarithm
contains a prime number.
75. log ( y ) + 5 log ( x ) = log ( 7 )
87. (a) ln ( 6 )
(b) log ( 28 )
88. (a) ln (10 )
(b) log ( 45 )
89. (a) log ( 60 )
(b) ln ( 270 )
90. (a) log (168)
(b) ln ( 480 )
76. ln ( y ) − 4 ln ( x ) = ln ( 5 )
77. ln ( y ) − 3ln ( x + 2 ) = 4 ln ( x ) − ln ( 7 )
78. log ( y ) + 5log ( 4 ) = 7 log ( x + 3) + 8 log ( x )
Use the Change of Base Formula to write the following in
terms of natural logarithms. Then use a calculator to
40
University of Houston Department of Mathematics
Exercise Set 3.3: Laws of Logarithms
If log b 2 = A, log b 3 = B , and log b 5 = C ( b > 1 ), write
each of the following logarithms in terms of A, B, and C.
91. (a) logb (10 )
1
(b) log b
3
92. (a) log b ( 6 )
3
(b) logb
2
2
93. (a) logb
5
(b) logb ( 25)
94. (a) log b ( 8 )
(b) log b ( 0.25 )
95. (a) logb ( 45)
27
(b) log b
20
1
96. (a) log b
90
(b) log b (150 )
97. (a) log b
( 12 )
3
(b) logb ( 0.09 )
(
98. (a) logb ( 0.3)
(b) log b
99. (a) log b (100b)
(b) log3 ( 5)
100. (a) log 5 ( 2 )
18
(b) logb
5b
MATH 1330 Precalculus
30
)
41
Exercise Set 3.4: Exponential and Logarithmic Equations and Inequalities
Solve each of the following equations.
16. 83 x− 2 = 5
(a) Solve for x. (Note: Further simplification may
occur in step (b).) Write the answer in terms
of natural logarithms, unless no logarithms
are involved.
x
17. 32 3 = 20
x
(b) Rewrite the answer so that each individual
logarithm from part (a) is written as a sum or
difference of logarithms of prime numbers,
(
)
( )
e.g., ln ( 40 ) = ln 23 ⋅ 5 = ln 23 + ln ( 5 )
= 3 ln ( 2 ) + ln ( 5 ) .
Write the answer in simplest form.
(c) If the simplified answer from part (b)
contains logarithms, rewrite the answer as a
decimal, correct to the nearest thousandth;
otherwise, simply rewrite the answer.
18. 12 6 = 18
19. 54 x = 67 x−1
20. 7 x + 3 = 158 x
21. 23 x −1 = 82 x +5
22. 124 x −3 = 45x + 7
23.
5
=7
3 − e− x
3
=4
2 − e− x
1.
7 x = 12
2.
8x = 3
24.
3.
e x = 22
25. e 2 x − 3e x − 40 = 0
4.
e x = 45
26. e 2 x + 5e x − 14 = 0
5.
3 ⋅ 6 x = 42
27. 32 x + 2 3x − 15 = 0
6.
2 ⋅ 9 x = 48
7.
( )
( )
28. 52 x − 4 5x − 21 = 0
x
e +7 =3
29. 49 x − 7 x − 6 = 0
8.
x
e − 9 = −3
30. 4 x + 8 ⋅ 2 x + 7 = 0
9.
19
−3 x
=7
31. x 2 ⋅ 7 x − 9 ⋅ 7 x = 0
10. 5
−2 x
=3
32. x 2 ⋅ 2 x − 16 ⋅ 2 x = 0
x
11. 6e = 13
33. x 2 e x − 4 xe x − 5e x = 0
x
12. 8e = 18
34. x 2 e x − 8 xe x + 12e x = 0
13. e
5x+4
− 5 = 31
35. 2log 2 (3 x −7) = 3 x − 7
14. 2e7 −3 x = −20
36. 6log6 (5 x +9) = 5 x + 9
15. 7
42
x+ 4
= 10
University of Houston Department of Mathematics
Exercise Set 3.4: Exponential and Logarithmic Equations and Inequalities
Solve each of the following equations. Write all
answers in simplest form.
56. log 2 ( x + 7) − log 2 ( x − 9) = log 2 11
37. ln x = 8
57. log5 ( 3x + 5 ) − log5 ( x + 11) = 0
38. ln x = 12
58. log 6 (2 x + 1) − log 6 ( x − 5) = 0
39. ln x = −7
59. log 2 ( x + 4 ) = 3 + log 2 ( x + 5)
40. ln x = −10
60. log3 (15 x + 8 ) = 2 + log 3 ( x + 1)
41. log x = 3
61. log 3 + log( x + 4) = log 5 + log( x − 3)
42. log x = −2
62. ln 6 + ln( x − 8) = ln 9 + ln( x + 3)
43. log5 ( x + 1) = 2
63. 2 log x = log 2 + log( x + 12)
44. log 2 (3x + 7) = 5
64. 2 log x = log 4 + log( x + 8)
45. log ( 4 x − 1) + 3 = 5
65. ln ln ( x ) = −5
46. log(2 x + 1) − 7 = −4
66. ln ln ( x ) = 5
47. ln(3x + 8) − 6 = 3
ln x
67. e ( ) = 5
48. ln(5 x − 4) + 2 = 9
ln x
68. e ( ) = −5
49. ln x + ln( x − 5) = ln 6
50. ln x + ln( x − 7) = ln18
( )
69. ln x 4 = 4 ln ( x )
4
51. log 2 x + log 2 ( x − 2) = 3
70. ln ( x ) = 4 ln ( x )
52. log 6 x + log 6 ( x + 1) = 1
71. log 7 3 x 2 = 2 log 7 ( 3 x )
53. log 2 ( x + 3) + log 2 ( x + 2) = 1
54. log 6 ( x + 2) + log 6 ( x + 3) = 1
( )
2
72. log 7 ( 3 x ) = 2 log 7 ( 3 x )
73. 2 log ( x ) = log 25
55. log 7 ( x + 8) − log 7 ( x − 5) = log 7 4
74. 2 ln ( x ) = ln (100 )
MATH 1330 Precalculus
43
Exercise Set 3.4: Exponential and Logarithmic Equations and Inequalities
For each of the following inequalities,
89. ln ( 7 − 3x ) ≤ 0
(a) Solve for x. (Note: Further simplification may
occur in step (b).) Write the answer in terms
of natural logarithms, unless no logarithms
are involved.
90. ln ( 5 x + 2 ) < 0
(b) Rewrite the answer so that each individual
logarithm from part (a) is written as a sum or
difference of logarithms of prime numbers,
(
)
( )
e.g., ln ( 40 ) = ln 23 ⋅ 5 = ln 23 + ln ( 5 )
= 3 ln ( 2 ) + ln ( 5 ) .
Simplify further if possible, and then write the
answer in interval notation.
(c) If the simplified answer from part (b)
contains logarithms, rewrite the answer as a
decimal, correct to the nearest thousandth;
otherwise, simply rewrite the answer. Then
write this answer in interval notation.
75. 8 x+ 3 > 15
76. 7 x− 4 ≤ 2
77. 7 x > 0
78. 7 x < 0
79. e 2 − x ≤ 9
80. e x + 4 > 10
81. 0.2 x < 7
82. 0.8x ≥ 6
(
)
83. 8 0.4 x − 2 > 109
(
)
84. 9 3 − 0.75 x ≤ 11
(
)
85. 5 8 + e x ≤ −6
86.
(9 − e ) ≤ 4
x
87. log 2 ( x ) ≥ 0
88. log 2 ( x ) < 0
44
University of Houston Department of Mathematics
Exercise Set 3.5: Applications of Exponential Functions
6.
Answer the following.
1.
Compound interest is calculated by the following
formula:
r
A(t ) = P 1 +
n
2.
(a) annually
(c) quarterly
(e) continuously
nt
State the meaning of each variable in the formula
above.
The present value of a sum of money is the
amount of money that must be invested now (in
the present) in order to obtain a certain amount
of money after t years. Find the present value of
$15,000 if interest is paid a rate of 8% per year,
compounded semiannually, for 5 years.
8.
The present value of a sum of money is the
amount of money that must be invested now (in
the present) in order to obtain a certain amount
of money after t years. Find the present value of
$20,000 if interest is paid a rate of 5% per year,
compounded quarterly, for 14 years.
9.
Find the annual percentage rate for an investment
that earns 6% per year, compounded quarterly.
A(t ) = Pe rt
3.
The present value of a sum of money is the
amount of money that must be invested now (in
the present) in order to obtain a certain amount
of money after t years.
In the following formula:
r
A(t ) = P 1 +
n
nt
(a) Which variable represents the present value?
(b) Which variable represents the amount of
money after t years?
4.
Exponential growth is calculated according to
the following formula:
N (t ) = N 0 ert
State the meaning of each variable in the formula
above.
Answer the following. In general, round answers to the
nearest hundredth. When giving answers for
population size, round to the nearest whole number.
5.
If $3,000 is invested at an interest rate of 7% per
year, find the value of the investment after 10
years if the interest is compounded:
(a) annually
(c) quarterly
(e) continuously
MATH 1330 Precalculus
(b) semiannually
(d) monthly
7.
Continuously compounded interest is calculated
by the following formula:
State the meaning of each variable in the formula
above.
If $18,000 is invested at an interest rate of 4%
per year, find the value of the investment after
15 years if the interest is compounded:
(b) semiannually
(d) monthly
10. Find the annual percentage rate for an investment
that earns 4% per year, compounded monthly.
11. Which of the following scenarios would be the
better investment?
(a) 7.5% per year, compounded quarterly
(b) 7.3% per year, compounded continuously
12. Which of the following scenarios would be the
better investment?
(a) 8% per year, compounded semiannually
(b) 7.97% per year, compounded continuously
13. A population of rabbits grows in such a way that
the population t days from now is given by
A(t ) = 250e0.02t .
(a) How many rabbits are present now?
(b) What is the relative growth rate of the
rabbits? (Write your answer as a
percentage.)
(c) How many rabbits will there be after 4 days?
(d) How many rabbits will there be after three
weeks?
(e) How many days will it take for the rabbit
population to double?
45
Exercise Set 3.5: Applications of Exponential Functions
14. A population of mosquitoes grows in such a way
that the population t hours from now is given by
A(t ) = 1,350e0.001t .
(a) How many mosquitoes are present now?
(b) What is the relative growth rate of the
mosquitoes? (Write your answer as a
percentage.)
(c) How many mosquitoes will there be after 6
hours?
(d) How many mosquitoes will there be after
two days?
(e) How many hours will it take for the
mosquito population to triple?
15. There are approximately 18,000 bacteria in a
culture, and the bacteria have a relative growth
rate of 75% per hour.
(a) Write a function that models the bacteria
population t hours from now.
(b) Use the function found in part (a) to estimate
the bacteria population 8 hours from now.
(c) In how many hours will the bacteria
population reach 99,000?
16. In the year 2000, there were approximately
35,000 people living in Exponentia, and the
population of Exponentia has a relative growth
rate of 4.1% per year.
(a) Write a function that models the population
of Exponentia t years from the year 2000.
(b) Use the function found in part (a) to estimate
the population of Exponentia in the year
2007.
(c) In how many years will the population reach
42,000?
17. The population of Smallville has a relative
growth rate of 1.2% per year. If the population of
Smallville in 1995 was 2,761, find the projected
population of the town in 2005.
18. The ladybug population in a certain area is
currently estimated to be 4,000, with a relative
growth rate of 1.5% per day. Estimate the
number of ladybugs 9 days from now.
19. An environmental group is studying a particular
type of buffalo, and they move a certain number
of these buffalo into a wildlife preserve. The
relative growth rate for this population of buffalo
is 2% per year. Five years after the buffalos are
moved into the preserve, the group finds that
there are 350 buffalo.
46
(a) How many buffalo were initially moved into
the preserve?
(b) Estimate the buffalo population after 10
years of being in the preserve.
(c) How long from the time of the initial move
will it take for the buffalo population to
grow to 1,000?
20. A certain culture of bacteria has a relative
growth rate of 135% per hour. Two hours after
the culture is formed, the count shows
approximately 10,000 bacteria.
(a) Find the initial number of bacteria in the
culture.
(b) Estimate the number of bacteria in the
culture 9 hours after the culture was started.
Round your answer to the nearest million.
(c) How long after the culture is formed will it
take for the bacteria population to grow to
1,000,000?
21. A certain radioactive substance decays according
to the formula m(t ) = 70e −0.062t , where m
represents the mass in grams that remains after t
days.
(a) Find the initial mass of the radioactive
substance.
(b) How much of the mass remains after 15
days?
(c) What percent of mass remains after 15 days?
(d) Find the half-life of this substance.
22. A certain radioactive substance decays according
to the formula m(t ) = 12e −0.089t , where m
represents the mass in grams that remains after t
days.
(a) Find the initial mass of the radioactive
substance.
(b) How much of the mass remains after 20
days?
(c) What percent of mass remains after 20 days?
(d) Find the half-life of this substance.
23. If $20,000 is invested at an interest rate of 5%
per year, compounded quarterly,
(a) Find the value of the investment after 7
years.
(b) How long will it take for the investment to
double in value?
University of Houston Department of Mathematics
Exercise Set 3.5: Applications of Exponential Functions
24. If $7,000 is invested at an interest rate of 6.5%
per year, compounded monthly,
(a) Find the value of the investment after 4
years.
(b) How long will it take for the investment to
triple in value?
25. If $4,000 is invested at an interest rate of 10%
per year, compounded continuously,
(a) Find the value of the investment after 15
years.
(b) How long will it take for the investment to
grow to a value of $7,000?
26. If $20,000 is invested at an interest rate of 5.25%
per year, compounded continuously,
(a) Find the value of the investment after 12
years.
(b) How long will it take for the investment to
grow to a value of $50,000?
27. A certain radioactive substance has a half-life of
50 years. If 75 grams are present initially,
(a) Write a function that models the mass of the
substance remaining after t years. (Within
the formula, round the rate to the nearest
ten-thousandth.)
(b) Use the function found in part (a) to estimate
how much of the mass remains after 80
years.
(c) How many years will it take for the
substance to decay to a mass of 10 grams?
28. A certain radioactive substance has a half-life of
120 years. If 850 milligrams are present initially,
(a) Write a function that models the mass of the
substance remaining after t years. (Within
the formula, round the rate to the nearest
ten-thousandth.)
(b) Use the function found in part (a) to estimate
how much of the mass remains after 500
years.
(c) How many years will it take for the
substance to decay to a mass of 200
milligrams?
MATH 1330 Precalculus
47
Exercise Set 4.1: Special Right Triangles and Trigonometric Ratios
Answer the following.
1.
2.
9.
4 2
If two sides of a triangle are congruent, then the
__________ opposite those sides are also
congruent.
If two angles of a triangle are congruent, then the
__________ opposite those angles are also
congruent.
45o
x
10.
45o
3.
In any triangle, the sum of the measures of its
angles is _____ degrees.
4.
In an isosceles right triangle, each acute angle
measures _____ degrees.
5.
Fill in each missing blank with one of the
following: smallest, largest
In any triangle, the longest side is opposite the
__________ angle, and the shortest side is
opposite the __________ angle.
3 2
x
11.
45o
8
x
6.
Fill in each missing blank with one of the
following: 30o, 60o, 90o
In a 30o-60o-90o triangle, the hypotenuse is
opposite the _____ angle, the shorter leg is
opposite the _____ angle, and the longer leg is
opposite the _____ angle.
For each of the following,
(a) Use the theorem for 45o-45o-90o triangles to
find x.
(b) Use the Pythagorean Theorem to verify the
result obtained in part (a).
12.
45o
7
x
13.
9
x
7.
45o
x
5
14.
12
x
8.
x
15.
45o
x
8
8 2
48
University of Houston Department of Mathematics
Exercise Set 4.1: Special Right Triangles and Trigonometric Ratios
22. In the figure below, an altitude is drawn to the
base of an equilateral triangle.
(a) Find a and b.
(b) Justify the answer obtained in part (a).
(c) Use the Pythagorean Theorem to find c, the
length of the altitude. (Write c in simplest
radical form.)
16.
45o
x
2 3
17.
2 3
30o 30o
4
c
45o
60o
x
60o
a
b
18.
x
For each of the following, Use the theorem for 30o-60o90o triangles to find x and y.
23.
5 2
30o
x
The following examples help to illustrate the theorem
regarding 30o-60o-90o triangles.
19. What is the measure of each angle of an
equilateral triangle?
y
7
24.
22
20. An altitude is drawn to the base of the equilateral
triangle drawn below. Find the measures of x and
y.
30
x
o
y
yo
25.
xo
21. In the figure below, an altitude is drawn to the
base of an equilateral triangle.
(a) Find a and b.
(b) Justify the answer obtained in part (a).
(c) Use the Pythagorean Theorem to find c, the
length of the altitude.
x
6 3
60o
y
26.
30o
x
y
30o 30o
10
c
60
o
a
8
60
o
b
MATH 1330 Precalculus
49
Exercise Set 4.1: Special Right Triangles and Trigonometric Ratios
27.
x
(b) Find the following:
5
60o
y
28.
y
60o
x
34.
sin ( A ) = _____
sin ( B ) = _____
cos ( A ) = _____
cos ( B ) = _____
tan ( A ) = _____
tan ( B ) = _____
D
25
15 3
E
F
24
29.
(a) Use the Pythagorean Theorem to find DE.
30o
x
(b) Find the following:
6
y
30.
x
sin ( F ) = _____
cos ( D ) = _____
cos ( F ) = _____
tan ( D ) = _____
tan ( F ) = _____
4 2
60o
35. Suppose that θ is an acute angle of a right
5
triangle and sin (θ ) = . Find cos (θ ) and
7
tan (θ ) .
y
31.
sin ( D ) = _____
y
60o
5 3
36. Suppose that θ is an acute angle of a right
triangle and tan (θ ) =
x
4 2
. Find sin (θ ) and
7
cos (θ ) .
32.
37. The reciprocal of the sine function is the
_______________ function.
y
x
30o
38. The reciprocal of the cosine function is the
_______________ function.
8
Answer the following. Write answers in simplest form.
33.
A
15
C
39. The reciprocal of the tangent function is the
_______________ function.
40. The reciprocal of the cosecant function is the
_______________ function.
17
41. The reciprocal of the secant function is the
_______________ function.
B
42. The reciprocal of the cotangent function is the
_______________ function.
(a) Use the Pythagorean Theorem to find BC.
50
University of Houston Department of Mathematics
Exercise Set 4.1: Special Right Triangles and Trigonometric Ratios
47. Suppose that θ is an acute angle of a right
43.
2 10
. Find the six
3
trigonometric functions of θ .
β
6
triangle and cot (θ ) =
5
α
x
(a) Use the Pythagorean Theorem to find x.
(b) Find the six trigonometric functions of α .
48. Suppose that θ is an acute angle of a right
5
triangle and sec (θ ) = . Find the six
2
trigonometric functions of θ .
(c) Find the six trigonometric functions of β .
49.
44.
α
60o
x
7
45o
3
β
4
(a) Use the Pythagorean Theorem to find x.
(b) Find the six trigonometric functions of α .
(c) Find the six trigonometric functions of β .
45.
α
30
2
o
x
4
β
8
(a) Use the Pythagorean Theorem to find x.
(a) Use the theorems for special right triangles
to find the missing side lengths in the
triangles above.
(b) Using the triangles above, find the
following:
sin ( 45 ) = _____
csc ( 45 ) = _____
cos ( 45 ) = _____
sec ( 45 ) = _____
tan ( 45 ) = _____
cot ( 45 ) = _____
(c) Using the triangles above, find the
following:
(b) Find the six trigonometric functions of α .
sin ( 30 ) = _____
csc ( 30 ) = _____
(c) Find the six trigonometric functions of β .
cos ( 30 ) = _____
sec ( 30 ) = _____
tan ( 30 ) = _____
cot ( 30 ) = _____
46.
β
8
6
α
x
(a) Use the Pythagorean Theorem to find x.
(b) Find the six trigonometric functions of α .
(c) Find the six trigonometric functions of β .
MATH 1330 Precalculus
(d) Using the triangles above, find the
following:
sin ( 60 ) = _____
csc ( 60 ) = _____
cos ( 60 ) = _____
sec ( 60 ) = _____
tan ( 60 ) = _____
cot ( 60 ) = _____
51
Exercise Set 4.1: Special Right Triangles and Trigonometric Ratios
50.
45o
o
12 30
8
60o
(a) Use the theorems for special right triangles
to find the missing side lengths in the
triangles above.
(b) Using the triangles above, find the
following:
sin ( 45 ) = _____
csc ( 45 ) = _____
cos ( 45 ) = _____
sec ( 45 ) = _____
tan ( 45 ) = _____
cot ( 45 ) = _____
(c) Using the triangles above, find the
following:
sin ( 30 ) = _____
csc ( 30 ) = _____
cos ( 30 ) = _____
sec ( 30 ) = _____
tan ( 30 ) = _____
cot ( 30 ) = _____
(d) Using the triangles above, find the
following:
sin ( 60 ) = _____
csc ( 60 ) = _____
cos ( 60 ) = _____
sec ( 60 ) = _____
tan ( 60 ) = _____
cot ( 60 ) = _____
51. Compare the answers to parts (b), (c), and (d) in
the previous two examples. What do you notice?
52
University of Houston Department of Mathematics
Exercise Set 4.2: Radians, Arc Length, and the Area of a Sector
If we use central angles of a circle to analyze angle
measure, a radian is an angle for which the arc of the
circle has the same length as the radius, as illustrated
in the figures below.
Answer the following.
7.
θ = 1 radian, since the arc
length is the same as the
length of the radius.
Note: Standard notation is
to say θ = 1 ; radians are
implied when there is no
angle measure.
r
θ
r
θ
(a) Use this figure as a guide to sketch and
estimate the number of radians in a complete
revolution.
2r
θ = 2 radians, since the arc
length is twice the length of
the radius.
Note: Standard notation is
to say θ = 2 .
θ
r
(b) Give an exact number for the number of
radians in a complete revolution. Justify
your answer. Then round this answer to the
nearest hundredth and compare it to the
result from part (a).
8.
28 cm
(b) 180 = _____ radians
Convert the following degree measures to radians.
First, give an exact result. Then round each answer to
the nearest hundredth.
9.
θ
Fill in the blanks:
(a) 360 = _____ radians
The number of radians can therefore be determined
s
by dividing the arc length s by the radius r, i.e. θ = .
r
Find θ in the examples below.
1.
In the figure below, θ = 1 radian.
(a) 30
(b) 90
(c) 135
10. (a) 45
(b) 60
(c) 150
11. (a) 120
(b) 225
(c) 330
12. (a) 210
(b) 270
(c) 315
13. (a) 19
(b) 40
(c) 72
14. (a) 10
(b) 53
(c) 88
7 cm
2.
30 ft
θ
Convert the following radian measures to degrees.
10 ft
(b)
4π
3
(c)
5π
6
(b)
7π
6
(c)
5π
4
17. (a) π
(b)
11π
12
(c)
61π
36
π
(b)
7π
18
(c)
53π
30
15. (a)
π
4
3.
r = 0.3 m; s = 72 cm
4.
r = 0.6 in; s = 3.06 in
5.
r = 64 ft; s = 32 ft
6.
r = 60 in; s = 21 ft
16. (a)
π
2
18. (a)
9
MATH 1330 Precalculus
53
Exercise Set 4.2: Radians, Arc Length, and the Area of a Sector
Convert the following radian measures to degrees.
Round answers to the nearest hundredth.
19. (a) 2.5
(b) 0.506
In numbers 31-34, change θ to radians and then find
the arc length using the formula s = rθ . Compare
results with those from exercises 23-26.
20. (a) 3.8
(b) 0.297
31. θ = 60 ; r = 12 cm
32. θ = 90 ; r = 10 in
Answer the following.
21. If two angles of a triangle have radian measures
π
2π
and
, find the radian measure of the third
12
5
angle.
22. If two angles of a triangle have radian measures
π
3π
and
, find the radian measure of the third
9
8
angle.
33. θ = 225 ; r = 4 ft
34. θ = 150 ; r = 12 cm
Find the missing measure in each example below.
s
35.
5π
6
9 in
To find the length of the arc of a circle, think of the
arc length as simply a fraction of the circumference of
the circle. If the central angle θ defining the arc is
given in degrees, then the arc length can be found
using the formula:
θ
s=
2π r )
(
360
36.
3m
80
s
Use the formula above to find the arc length s.
23. θ = 60 ; r = 12 cm
24. θ = 90 ; r = 10 in
25. θ = 225 ; r = 4 ft
37. r = 6 cm; θ = 300 ; s = ?
38. r = 10 ft; θ =
26. θ = 150 ; r = 12 cm
If the central angle θ defining the arc is instead given
in radians, then the arc length can be found using the
formula:
s=
θ
2π
( 2π r ) = rθ
Use the formula s = rθ to find the arc length s:
27. θ =
7π
; r = 9 yd
6
28. θ =
3π
; r = 6 cm
4
29. θ = π ; r = 2 ft
30. θ =
54
5π
; r = 30 in
3
3π
; s= ?
4
39. s =
12π
π
m; θ = ; r = ?
5
2
40. s =
50π
5π
in; θ =
; r= ?
3
6
41. s = 7 ft; θ =
3π
; r= ?
4
2π
; r= ?
3
43. s = 12 in; θ = 5; r = ?
42. s = 20 cm; θ =
44. s = 7 in; θ = 3; r = ?
45. s = 15 m; θ = 270 ; r = ?
46. s = 8 yd; θ = 225 ; r = ?
University of Houston Department of Mathematics
Exercise Set 4.2: Radians, Arc Length, and the Area of a Sector
Answer the following.
Answer the following.
47. Find the perimeter of a sector of a circle with
π
central angle
and radius 8 cm.
6
48. Find the perimeter of a sector of a circle with
7π
central angle
and radius 3 ft.
4
61. A sector of a circle has central angle
360
Use the formula above to find the area of the sector:
62. A sector of a circle has central angle
51. θ = 225 ; r = 4 ft
52. θ = 150 ; r = 12 cm
If the central angle θ defining the sector is instead
given in radians, then the area of the sector can be
found using the formula:
θ
A=
( π r 2 ) = 12 r 2θ
2π
and area
5π
and
6
5π 2
ft . Find the radius of the circle.
27
63. A sector of a circle has central angle 120 and
16π 2
area
in . Find the radius of the circle.
75
64. A sector of a circle has central angle 210 and
21π 2
m . Find the radius of the circle.
area
4
49. θ = 60 ; r = 12 cm
50. θ = 90 ; r = 10 in
4
49π
cm 2 . Find the radius of the circle.
8
area
To find the area of a sector of a circle, think of the
sector as simply a fraction of the circle. If the central
angle θ defining the sector is given in degrees, then
the area of the sector can be found using the formula:
θ
A=
π r2 )
(
π
65. A sector of a circle has radius 6 ft and area
63π 2
ft . Find the central angle of the sector (in
2
radians).
66. A sector of a circle has radius
4
cm and area
7
8π
cm 2 . Find the central angle of the sector (in
49
radians).
2
Use the formula A = 12 r θ to find the area of the
sector:
53. θ =
7π
; r = 9 yd
6
54. θ =
3π
; r = 6 cm
4
55. θ = π ; r = 2 ft
56. θ =
5π
; r = 30 in
3
In numbers 57-60, change θ to radians and then find
the area of the sector using the formula A = 12 r 2θ .
Compare results with those from exercises 49-52.
57. θ = 60 ; r = 12 cm
58. θ = 90 ; r = 10 in
59. θ = 225 ; r = 4 ft
60. θ = 150 ; r = 12 cm
MATH 1330 Precalculus
Answer the following. SHOW ALL WORK involved
in obtaining each answer. Give exact answers unless
otherwise indicated.
67. A CD has a radius of 6 cm. If the CD’s rate of
turn is 900o/sec, find the following.
(a) The angular speed in units of radians/sec.
(b) The linear speed in units of cm/sec of a
point on the outer edge of the CD.
(c) The linear speed in units of cm/sec of a
point halfway between the center of the CD
and its outer edge.
68. Each blade of a fan has a radius of 11 inches. If
the fan’s rate of turn is 1440o/sec, find the
following.
(a) The angular speed in units of radians/sec.
(b) The linear speed in units of inches/sec of a
point on the outer edge of the blade.
(c) The linear speed in units of inches/sec of a
point on the blade 7 inches from the center.
55
Exercise Set 4.2: Radians, Arc Length, and the Area of a Sector
69. A bicycle has wheels measuring 26 inches in
diameter. If the bicycle is traveling at a rate of 20
miles per hour, find the wheels’ rate of turn in
revolutions per minute (rpm). Round the answer
to the nearest hundredth.
70. A car has wheels measuring 16 inches in
diameter. If the car is traveling at a rate of 55
miles per hour, find the wheels’ rate of turn in
revolutions per minute (rpm). Round the answer
to the nearest hundredth.
71. A car has wheels with a 10 inch radius. If each
wheel’s rate of turn is 4 revolutions per second,
(a) Find the angular speed in units of
radians/second.
(b) How fast is the car moving in units of
inches/sec?
(c) How fast is the car moving in miles per
hour? Round the answer to the nearest
hundredth.
72. A bicycle has wheels with a 12 inch radius. If
each wheel’s rate of turn is 2 revolutions per
second,
(a) Find the angular speed in units of
radians/second.
(b) How fast is the bicycle moving in units of
inches/sec?
(c) How fast is the bicycle moving in miles per
hour? Round the answer to the nearest
hundredth.
73. A clock has an hour hand, minute hand, and
second hand that measure 4 inches, 5 inches, and
6 inches, respectively. Find the distance traveled
by the tip of each hand in 20 minutes.
74. An outdoor clock has an hour hand, minute hand,
and second hand that measure 12 inches, 14
inches, and 15 inches, respectively. Find the
distance traveled by the tip of each hand in 45
minutes.
56
University of Houston Department of Mathematics
Exercise Set 4.3: Unit Circle Trigonometry
Sketch each of the following angles in standard position.
(Do not use a protractor; just draw a quick sketch of each
angle.)
1.
(a) 30
2.
(a) 120
(b) −135
(c) 300
(b) −60
(c) 210
4
4π
(b)
3
11π
(c)
6
5.
(a) 90
(b) −π
(c) 450
6.
(a) 270
(b) 180
(c) −4π
7.
(a) −240
(b)
8.
(a) −315
(b) −
3
4.
(a) −
π
(a) 50
10. (a) 300
11. (a)
12. (a)
(c) 60
18. (a) 315
(b) 150
(c) −120
13π
6
(c) −510
9π
4
(c) 1020
19. (a)
7π
6
20. (a) −
π
22. (a)
(b)
5π
3
(c) −
7π
4
(b)
5π
4
(c) −
5π
6
3
(b) −
21. (a) 840
11π
3
(b) −780
8π
3
(b) −
(c) −
11π
4
23. Using the following unit circle, draw and then label
π
the terminal side of all multiples of
from 0 to
(b) −830
(b) −
(c) −660
2
2π radians. Write all labels in simplest form.
(b) −200
2π
5
37π
6
The exercises below are helpful in creating a
comprehensive diagram of the unit circle. Answer the
following.
Find three angles, one negative and two positive, that are
coterminal with each angle below.
9.
(b) −30
5π
(c) −
6
(a)
17. (a) 240
7π
(b)
4
3.
π
Sketch each of the following angles in standard position
and then specify the reference angle or reference number.
y
7π
2
1
4π
9
0
1
-1
x
-1
Answer the following.
13. List four quadrantal angles in degree measure,
where each angle θ satisfies the condition
0 ≤ θ < 360 .
24. Using the following unit circle, draw and then label
π
the terminal side of all multiples of
from 0 to
14. List four quadrantal angles in radian measure, where
π
5π
each angle θ satisfies the condition ≤ θ <
.
2
4
2π radians. Write all labels in simplest form.
2
y
15. List four quadrantal angles in radian measure, where
5π
13π
each angle θ satisfies the condition
<θ ≤
.
4
16. List four quadrantal angles in degree measure,
where each angle θ satisfies the condition
800 < θ ≤ 1160 .
1
4
0
1
-1
x
-1
MATH 1330 Precalculus
57
Exercise Set 4.3: Unit Circle Trigonometry
25. Using the following unit circle, draw and then label
π
the terminal side of all multiples of
from 0 to
28. Label all the special angles on the unit circle in
degrees.
3
2π radians. Write all labels in simplest form.
y
y
1
1
0
1
-1
x
1
-1
x
-1
-1
26. Using the following unit circle, draw and then label
π
the terminal side of all multiples of
from 0 to
6
2π radians. Write all labels in simplest form.
Name the quadrant in which the given conditions are
satisfied.
y
29. sin (θ ) > 0, cos (θ ) < 0
1
30. sin (θ ) < 0, sec (θ ) > 0
0
1
-1
31. cot (θ ) > 0, sec (θ ) < 0
x
32. csc (θ ) > 0, cot (θ ) > 0
-1
33. tan (θ ) < 0, csc (θ ) < 0
27. Use the information from numbers 23-26 to label all
the special angles on the unit circle in radians.
34. csc (θ ) < 0, tan (θ ) > 0
y
Fill in each blank with < , > , or = .
( )
(
)
( )
(
)
35. sin 40 _____ sin 140
1
36. cos 20 _____ cos 160
1
-1
x
(
)
( )
(
)
( )
(
)
(
)
(
)
(
)
37. tan 310 _____ tan 50
38. sin 195 _____ sin 15
-1
39. cos 355 _____ cos 185
40. tan 110 _____ tan 290
58
University of Houston Department of Mathematics
Exercise Set 4.3: Unit Circle Trigonometry
Let P ( x , y ) denote the point where the terminal side of
an angle θ meets the unit circle. Use the given
information to evaluate the six trigonometric functions of
θ.
41. P is in Quadrant I and y =
57. (a) csc
25π
6
(b) cot ( −460 )
58. (a) tan ( 520 )
9π
(b) sec −
4
2
.
3
3
8
42. P is in Quadrant I and x = .
43. P is in Quadrant IV and x =
4
.
5
44. P is in Quadrant III and y = −
24
.
25
1
5
45. P is in Quadrant II and x = − .
2
46. P is in Quadrant II and y = .
7
For each quadrantal angle below, give the coordinates of
the point where the terminal side of the angle intersects
the unit circle. Then give the six trigonometric functions
of the angle. If a value is undefined, state “Undefined.”
For each angle below, give the coordinates of the point
where the terminal side of the angle intersects the unit
circle. Then give the six trigonometric functions of the
angle.
59. 45
60. 60
61. 210
62. 135
63.
5π
3
64.
11π
6
An alternate method of finding trigonometric functions of
30o, 45o, or 60o is shown below.
47. 90
48. −180
65.
10
49. −2π
3π
50.
2
5π
51. −
2
Rewrite each expression in terms of its reference angle,
deciding on the appropriate sign (positive or negative).
For example,
)
4π
π
tan
= tan
3
3
7
π
π
cos
= − cos
6
6
( )
sin 240 = − sin 60
(
60o
30o
30o
Diagram 1
Diagram 2
3
(a) Find the missing side measures in each of the
diagrams above.
52. 8π
(
60o
)
( )
sec −315 = sec 45
(b) Use right triangle trigonometric ratios to find
the following, using Diagram 1:
sin ( 30 ) = _____
sin ( 60 ) = _____
cos ( 30 ) = _____
cos ( 60 ) = _____
tan ( 30 ) = _____
tan ( 60 ) = _____
(c) Repeat part (b), using Diagram 2.
53. (a) cos ( 300 )
(b) tan (135 )
54. (a) sin ( −45 )
(b) cot ( 210 )
55. (a) sin (140 )
56. (a) csc ( −190 )
2π
3
11π
(b) cos
6
(b) sec −
MATH 1330 Precalculus
(d) Use the unit circle to find the trigonometric
ratios listed in part (b).
(e) Examine the answers in parts (b) through (d).
What do you notice?
59
Exercise Set 4.3: Unit Circle Trigonometry
66.
45o
45o
Use either the unit circle or the right triangle method
from numbers 65-70 to evaluate the following. (Note: The
right triangle method can not be used for quadrantal
angles.) If a value is undefined, state “Undefined.”
45o
8
45o
7
71. (a) tan ( 30 )
(b) sin ( −135 )
72. (a) cos (180 )
(b) csc ( −60 )
73. (a) csc ( −150 )
(b) sin ( 270 )
74. (a) sec ( 225 )
(b) tan ( −240 )
75. (a) cot ( −450 )
(b) cos ( 495 )
76. (a) sin ( −210 )
(b) cot ( −420 )
77. (a) csc (π )
2π
(b) cos
3
(d) Use the unit circle to find the trigonometric
ratios listed in part (b).
π
78. (a) sin −
3
5π
(b) cot
4
(e) Examine the answers in parts (b) through (d).
What do you notice?
π
79. (a) sec −
6
(b) tan
Diagram 1
Diagram 2
(a) Find the missing side measures in each of the
diagrams above.
(b) Use right triangle trigonometric ratios to find
the following, using Diagram 1:
sin ( 45 ) = _____
csc ( 45 ) = _____
cos ( 45 ) = _____
sec ( 60 ) = _____
tan ( 45 ) = _____
cot ( 60 ) = _____
(c) Repeat part (b), using Diagram 2.
11π
4
The following two diagrams can be used to quickly
evaluate the trigonometric functions of any angle having
a reference angle of 30o, 45o, or 60o. Use right
trigonometric ratios along with the concept of reference
angles, to evaluate the following. (Remember that when
π
π
π
converting degrees to radians, 30 = , 45 = , 60 = .)
6
2
60o
4
45o
2
1
45o
30o
1
3
67. (a) sin ( 240
)
(b) tan (135
)
(b) csc ( −225 )
π
69. (a) cos −
4
(b) sec
2π
70. (a) sin
3
7π
(b) cot −
6
(b) sec −
10π
81. (a) cot −
3
(b)
82. (a) tan ( −5π )
11π
(b) cos −
2
7π
sec
4
3
Use a calculator to evaluate the following to the nearest
ten-thousandth. Make sure that your calculator is in the
appropriate mode (degrees or radians).
11π
6
For example, when evaluating csc (θ ) on your calculator, use the identity
csc (θ ) =
1 . Do NOT use the calculator key labeled sin −1 (θ ) ; this
sin (θ )
represents the inverse sine function, which will be discussed in Section 5.4.
( )
tan ( 350 )
csc (191 )
83. (a) sin 37
84. (a)
85. (a)
60
3π
2
80. (a) csc
Note: Be careful when evaluating the reciprocal trigonometric functions.
68. (a) cos ( 330 )
1
3π
4
( )
cos ( −84 )
cot ( 21 )
(b) tan −218
(b)
(b)
University of Houston Department of Mathematics
Exercise Set 4.3: Unit Circle Trigonometry
(
86. (a) cot 310
)
( )
(b) sec 73
π
87. (a) cos
5
11π
(b) csc −
7
10π
88. (a) sin −
9
(b) cot ( 4.7π )
89. (a) tan ( −4.5 )
(b) sec ( 3)
90. (a) csc ( −0.457 )
(b) tan ( 9.4 )
MATH 1330 Precalculus
61
Exercise Set 4.4: Trigonometric Expressions and Identities
Answer the following.
1.
Beginning with the Pythagorean identity
cos 2 (θ ) + sin 2 (θ ) = 1 , establish another
Pythagorean identity by dividing each term by
cos 2 (θ ) . Show all work.
2.
Beginning with the Pythagorean identity
cos 2 (θ ) + sin 2 (θ ) = 1 , establish another
Pythagorean identity by dividing each term by
sin 2 (θ ) . Show all work.
Solve the following algebraically, using identities from
this section.
3.
4.
5.
6.
7.
8.
9.
12
π
and < θ < π , find the five
2
5
remaining trigonometric functions of θ .
12. If cot (θ ) = −
5
3π
and
< θ < 2π , find the exact
2
13
values of sin (θ ) and tan (θ ) .
Perform the following operations and combine like
terms. (Do not rewrite the terms or the solution in
terms of any other trigonometric functions.)
13. sin (θ ) − 3 sin (θ ) + 5
14. tan (θ ) − 7 tan (θ ) − 2
15. csc ( x ) − 3
2
16. 3 + sec ( x )
2
If cos (θ ) =
4
π
and < θ < π , find the exact
2
5
values of cos (θ ) and tan (θ ) .
If sin (θ ) =
1
If sin (θ ) = − and 180 < θ < 270 , find the
9
five remaining trigonometric functions of θ .
1
and 270 < θ < 360 , find the
10
five remaining trigonometric functions of θ .
If cos (θ ) =
17.
3
7
−
cos (θ ) sin (θ )
18. −
6
2
+
sin (θ ) cos (θ )
19. 5 tan (θ ) sin (θ ) − 8 tan (θ ) sin (θ )
20. 8 cos (θ ) cot (θ ) + 9 cos (θ ) cot (θ )
Factor each of the following expressions.
21. 25sin 2 (θ ) − 49 cos 2 (θ )
9
3π
and
< θ < 2π , find the exact
4
2
values of cot (θ ) and sin (θ ) .
22. 16 cos 2 (θ ) − 81sin 2 (θ )
5
3π
and π < θ <
, find the exact
2
2
values of tan (θ ) and cos (θ ) .
24. tan 2 (θ ) + 9 tan (θ ) + 12
13
and 90 < θ < 180 , find the
6
exact values of csc (θ ) and sin (θ ) .
26. 6 csc 2 (θ ) + 7 csc (θ ) − 5
If csc (θ ) = −
If sec (θ ) = −
If cot (θ ) = −
23. sec2 (θ ) − 7 sec (θ ) + 12
25. 10 cot 2 (θ ) − 13cot (θ ) − 3
Simplify the following.
1
10. If tan (θ ) = and 180 < θ < 270 , find the
3
exact values of sec (θ ) and cot (θ ) .
3
3π
and π < θ <
, find the five
4
2
remaining trigonometric functions of θ .
11. If tan (θ ) =
62
27. cos (θ ) tan (θ )
28. sin (θ ) cot (θ )
29. csc 2 ( x ) tan 2 ( x )
30. sec3 ( x ) cot 3 ( x )
University of Houston Department of Mathematics
Exercise Set 4.4: Trigonometric Expressions and Identities
31. cot ( x ) csc ( x ) tan 2 ( x )
32. sin ( x ) sec ( x ) cot
2
( x)
33. 1 − sin 2 ( x )
34. cos
2
52.
sec4 ( x ) − tan 4 ( x )
sec 2 ( x ) + tan 2 ( x )
Prove each of the following identities.
( x) −1
53. sec ( x ) − sin ( x ) tan ( x ) = cos ( x )
35. sec (θ ) − 1 sec (θ ) + 1
36. csc ( x ) + 1 csc ( x ) − 1
54. cos ( x ) cot ( x ) − csc ( x ) = − sin ( x )
55.
37. sin (θ ) csc (θ ) − sin (θ )
sec (θ ) csc (θ )
tan (θ ) + cot (θ )
=1
2
38. cos (θ ) sec (θ ) − cos (θ )
39. sec 2 (θ ) − 1 csc 2 (θ ) − 1
40.
57. tan ( x ) − sin ( x ) cos ( x ) = tan ( x ) sin 2 ( x )
58. cot ( x ) − cos ( x ) sin ( x ) = cot ( x ) cos 2 ( x )
1 − sin 2 ( x )
cos 2 ( x ) − 1
59. cot 2 ( x ) − cos 2 ( x ) = cot 2 ( x ) cos 2 ( x )
41. sin ( x ) cot ( x ) + tan ( x )
60. tan 2 ( x ) − sin 2 ( x ) = tan 2 ( x ) sin 2 ( x )
42. cos ( x ) tan ( x ) + cot ( x )
61.
43. sec ( x ) − tan ( x ) sec ( x ) + tan ( x )
44. csc ( x ) − cot ( x ) csc ( x ) + cot ( x )
62.
45. 1 − cos ( x ) csc ( x ) + cot ( x )
46. csc ( x ) − 1 sec ( x ) + tan ( x )
47.
48.
cot ( x ) + tan ( x )
tan ( x )
1 + sec ( x )
+
= csc 2 ( x ) − sec 2 ( x )
1 + sec ( x )
tan ( x )
= 2 csc ( x )
csc 2 ( x )
65. tan 4 (θ ) + tan 2 (θ ) = sec4 (θ ) − sec 2 (θ )
sec2 ( x )
66. csc 4 ( x ) − csc2 ( x ) = cot 4 ( x ) + cot 2 ( x )
tan ( x ) + cot ( x )
cos ( x )
cos ( x )
−
1 − sin ( x ) 1 + sin ( x )
cot (θ )
cot (θ )
+
csc (θ ) + 1 csc (θ ) − 1
69.
51.
sin ( x ) cos ( x )
64. 3sin 2 ( x ) − 4 cos 2 ( x ) = 3 − 7 cos 2 ( x )
68.
50.
cot ( x ) − tan ( x )
63. 2 cos 2 ( x ) + 5sin 2 ( x ) = 2 + 3sin 2 ( x )
67.
49.
2
56. sin (θ ) − cos (θ ) + sin (θ ) + cos (θ ) = 2
2
2
4
4
cos (θ ) − sin (θ )
sin (θ ) − cos (θ )
70.
MATH 1330 Precalculus
sin ( x )
1 + cos ( x )
cos ( x )
1 + sin ( x )
cos ( x )
1 + sin ( x )
1 − cos (θ )
sin (θ )
= csc ( x ) − cot ( x )
= sec ( x ) − tan ( x )
=
=
1 − sin ( x )
cos ( x )
sin (θ )
1 + cos (θ )
63
Exercise Set 4.4: Trigonometric Expressions and Identities
Another method of solving problems like exercises 312 is shown below.
Let the terminal side of an angle θ intersect the circle x 2 + y 2 = r 2 .
First draw the reference angle, α , for θ . Then, draw a right triangle
with its hypotenuse extending from the origin to ( x, y ) and its leg is on
the x-axis, as shown below. Label the legs x and y (which may be
negative), and the hypotenuse with positive length r.
y
Answer the following.
77. If the terminal ray of an angle θ in standard
position passes through the point (12, − 5 ) , find
the six trigonometric functions of θ .
78. If the terminal ray of an angle θ in standard
position passes through the point ( −8, 15 ) , find
the six trigonometric functions of θ .
( x, y )
79. If the terminal ray of an angle θ in standard
position passes through the point ( −6, − 2 ) , find
r
the six trigonometric functions of θ .
y
α
θ
x
x
80. If the terminal ray of an angle θ in standard
position passes through the point ( 3, − 9 ) , find
the six trigonometric functions of θ .
Then find the six trigonometric functions of α , using right triangle
the leg opposite angle α
, etc…) The six
the hypotenuse
trigonometric functions of α are the same as the six trigonometric
y
functions of θ . (This can be compared with the ratios sin θ = , etc. at
r
the beginning of the text in Section 4.3).
trigonometry ( sin α =
In each of the following examples, use the given
information to sketch and label a right triangle in the
appropriate quadrant, as described above. Use the
Pythagorean Theorem to find the missing side, and
then use right triangle trigonometry to evaluate the
five remaining trigonometric functions of θ .
71. sin (θ ) = −
72. cos (θ ) = −
73. sec (θ ) = −
1
and 270 < θ < 360
4
13
and 180 < θ < 270
7
7
π
and < θ < π
2
2
74. csc (θ ) = 5 and
75. tan (θ ) =
2
<θ <π
2
3π
and π < θ <
2
5
76. cot (θ ) = −
64
π
11
3π
and
< θ < 2π
2
5
University of Houston Department of Mathematics
Exercise Set 5.1: Trigonometric Functions of Real Numbers
5π
14. (a) csc −
4
5π
(b) tan −
3
Note: These questions are similar to those in Chapter 4.
The difference is that these are trigonometric functions of
real numbers, rather than of angles. The solution process,
however, is the same.
11π
15. (a) sin
4
8π
(b) cot
3
17π
16. (a) cos
6
15π
(b) tan
4
Use the given information to evaluate the five
remaining trigonometric functions of t. If a value is
undefined, state “Undefined.”
1.
sin ( t ) = −
2
3π
and
< t < 2π
2
7
11π
17. (a) cot
2
13π
(b) sec
3
2.
cos ( t ) = −
1
3π
and π < t <
2
5
31π
18. (a) csc
6
13π
(b) sec
2
3.
sec ( t ) = −4 and
19. (a) csc ( 9π )
19π
(b) cos −
3
35π
20. (a) tan −
4
(b) sin ( −11π )
π
2
<t <π
8
3π
and
< t < 2π
2
5
4.
csc ( t ) = −
5.
cot ( t ) =
6.
tan ( t ) = −
15
π
3π
and < t <
2
2
8
7.
tan ( t ) = −
2 10
3π
and
< t < 2π
2
3
8.
cot ( t ) = 4 3 and π < t <
4
and π < t < 2π
3
3π
2
Use the opposite-angle identities and/or the periodicity
identities to evaluate the following.
Note: These questions are similar to those in Chapter 4.
The identities mentioned above offer an alternative
method of finding trigonometric functions of negative
numbers or of numbers whose absolute value is greater
than 2π .
π
(a) cos −
4
π
(b) csc −
6
π
10. (a) sin −
3
π
(b) sec −
4
2π
11. (a) sin −
3
π
(b) tan −
2
π
12. (a) cos −
2
5π
(b) cot −
6
4π
13. (a) sec −
3
11π
(b) cot −
6
9.
MATH 1330 Precalculus
49π
sin
4
21. (a)
29π
tan
4
14π
17π
(b) tan ( 6π ) − sin
cos
3
6
19π
csc
3
22. (a)
19π
sec
3
10π
(b) cot
3
16π
cot
3
+
cos 17π
6
20π 11π
sin
cot
3 4
23. (a)
5π
cos −
6
19π 31π
sin
sin
3 6
(b)
5π
cos ( 8π ) tan
4
65
Exercise Set 5.1: Trigonometric Functions of Real Numbers
11π
19π
tan
cos
4
6
24. (a)
31π
sin
3
3π
cos sin ( 6π )
2
(b)
19
π
21π
sin
tan
2
4
Simplify the following. Write all trigonometric
functions in terms of t.
25. sin ( −t ) csc ( t )
Prove each of the following identities.
39.
40.
41.
30.
tan ( t )
1 + cos ( −t )
sin ( t )
cos ( −t ) − 1
43.
1 + sec ( −t )
44.
1 − csc ( −t )
45.
sin ( −t )
cot ( −t )
cos ( −t )
sin ( −t )
1 − sin ( −t )
28. csc ( −t ) tan ( −t )
29.
1 − sin ( −t )
42.
26. cos ( −t ) sec ( t )
27. sec ( −t ) cot ( −t )
cos ( −t )
46.
cos ( −t )
csc ( −t )
sec ( −t )
= sec ( −t ) + tan ( −t )
= cot ( t ) + csc ( −t )
=
=
1 + cos ( −t )
sin ( −t )
cos ( t )
1 + sin ( −t )
= sin ( −t ) + tan ( −t )
= cos ( −t ) − cot ( −t )
sin ( t − 4π )
1 + cos ( t + 2π )
sin ( t + 8π )
1 − cot ( t − 2π )
+
−
1 + cos ( t − 2π )
sin ( t + 6π )
cos ( t − 2π )
tan ( t + π ) − 1
= 2 csc ( t + 4π )
= sin ( t ) + cos ( t )
31. cos ( −t ) cot ( −t ) + sin ( −t )
32. sin ( −t ) tan ( −t ) + cos ( −t )
33. sin ( −t ) + sin ( −t ) cot 2 ( −t )
34. cos ( −t ) + cos ( −t ) tan 2 ( −t )
35. sec ( t + 4π ) cos ( t + 2π )
36. sin ( t + 6π ) csc ( t − 2π )
37.
38.
66
1 + tan ( t − π )
1 + cot ( t + 2π )
sec ( t + 2π ) + csc ( t − 6π )
1 + tan ( t + 3π )
University of Houston Department of Mathematics
Exercise Set 5.2: Graphs of the Sine and Cosine Functions
For each of the following functions,
(a) Find the period.
(b) Find the amplitude.
6.
8 y
6
4
2
1.
x
4 y
−π
3
−2π/3
−π/3
π/3
2π/3
−2
2
−4
1
x
−6
−4
−2
2
−1
4
6
8
10
12
Answer the following.
−2
7.
2.
6 y
(a) Use your calculator to complete the
following chart. Round to the nearest
hundredth.
4
y = sin ( x )
x
2
x
y = sin ( x )
x
−6
−4
0
π
−2
π
−4
6
7π
6
−6
π
−2
2
4
6
8
5π
4
4
π
3.
6 y
3
4
π
2
x
−2
−1
1
2
−2
−4
−6
4π
3
2
3π
2
2π
3
5π
3
3π
4
7π
4
5π
6
11π
6
2π
4.
2
y
x
−1
1
2
−2
(b) Plot the points from part (a) to discover the
graph of f ( x ) = sin ( x ) .
1.2
−4
y
1.0
0.8
−6
0.6
0.4
−8
0.2
−0.2
x
π/6
π/3
π/2
2π/3 5π/6
π
7π/6 4π/3 3π/2 5π/3 11π/6 2π
−0.4
5.
2
−π/2
−π/4
y
−0.6
−0.8
π/4
π/2
3π/4
π
x
−1.0
5π/4
−1.2
−2
−4
−6
−8
MATH 1330 Precalculus
(c) Use the graph from part (b) to sketch an
extended graph of f ( x ) = sin ( x ) , where
−4π ≤ x ≤ 4π . Be sure to show the
intercepts as well as the maximum and
minimum values of the function.
67
Exercise Set 5.2: Graphs of the Sine and Cosine Functions
(d) State the domain and range of
f ( x ) = sin ( x ) . Do not base this answer
(c) Use the graph from part (b) to sketch an
extended graph of g ( x ) = cos ( x ) , where
−4π ≤ x ≤ 4π . Be sure to show the
intercepts as well as the maximum and
minimum values of the function.
simply on the limited domain from part (c),
but the entire graph of f ( x ) = sin ( x ) .
(e) State the amplitude and period of
f ( x ) = sin ( x ) .
(d) State the domain and range of
g ( x ) = cos ( x ) . Do not base this answer
simply on the limited domain from part (c),
but the entire graph of g ( x ) = cos ( x ) .
(f) Give an interval on which f ( x ) = sin ( x ) is
increasing.
8.
(e) State the amplitude and period of
g ( x ) = cos ( x ) .
(a) Use your calculator to complete the
following chart. Round to the nearest
hundredth.
y = cos ( x )
x
x
0
π
π
7π
6
(f) Give an interval on which g ( x ) = cos ( x ) is
decreasing.
y = cos ( x )
Answer the following.
6
π
9.
5π
4
4
π
determine the coordinates of points A, B, C, and
D.
4π
3
3
π
2
3π
2
2π
3
5π
3
3π
4
7π
4
5π
6
11π
6
Use the graph of f ( x ) = cos ( x ) below to
y
f
C
D
A
x
B
2π
10. Use the graph of f ( x ) = sin ( x ) below to
(b) Plot the points from part (a) to discover the
graph of g ( x ) = cos ( x ) .
1.2
determine the coordinates of points A, B, C, and
D.
y
1.0
y
0.8
f
0.6
0.4
0.2
−0.2
x
π/6
π/3
π/2
2π/3 5π/6
π
C
7π/6 4π/3 3π/2 5π/3 11π/6 2π
D
x
−0.4
−0.6
−0.8
−1.0
−1.2
68
A
B
University of Houston Department of Mathematics
Exercise Set 5.2: Graphs of the Sine and Cosine Functions
Use the graphs of f ( x ) = sin ( x ) and g ( x ) = cos ( x )
to answer the following.
16. For −2π ≤ x < 0 , use the graph of
f ( x ) = sin ( x ) to find the value(s) of x for
which:
11. Use the appropriate graph to find the following
function values.
(a) sin ( x ) = 1
(a) cos ( 0 )
(b) sin ( x ) = −1
3π
(b) cos −
2
(c) sin ( x ) = 0
5π
(c) sin
2
(d) sin ( −3π )
12. Use the appropriate graph to find the following
function values.
Graph each of the following functions over the interval
−2π ≤ x ≤ 2π .
17. f ( x ) = cos ( x )
18. g ( x ) = sin ( x )
(a) sin ( 0 )
π
(b) sin −
2
(c) cos (π )
7π
(d) cos −
2
Matching. The left-hand column contains equations
that represent transformations of f ( x ) = sin ( x ) .
Match the equations on the left with the description on
the right of how to obtain the graph of y = g ( x ) from
the graph of f .
19. y = sin ( x − 3)
13. For 0 ≤ x < 2π , use the graph of f ( x ) = sin ( x )
to find the value(s) of x for which:
(a) sin ( x ) = 1
(b) sin ( x ) = 0
(c) sin ( x ) = −1
14. For 0 ≤ x < 2π , use the graph of g ( x ) = cos ( x )
to find the value(s) of x for which:
(a) cos ( x ) = 0
(b) cos ( x ) = −1
(c) cos ( x ) = 1
15. For −2π ≤ x < 0 , use the graph of
g ( x ) = cos ( x ) to find the value(s) of x for
which:
(a) cos ( x ) = −1
(b) cos ( x ) = 1
(c) cos ( x ) = 0
MATH 1330 Precalculus
A. Shrink horizontally by a
1
factor of 3 .
20. y = sin ( x ) − 3
21. y = sin ( 3 x )
B. Shift left 3 units, then
reflect in the y-axis.
1
22. y = sin ( x )
3
C. Reflect in the x-axis,
then shift downward 3
units.
23. y = 3sin ( x )
D. Shift right 3 units.
24. y = sin ( x + 3 )
E. Shift right 2 units, then
reflect in the x-axis, then
shift upward 3 units.
25. y = sin ( − x + 3 )
F. Shift upward 3 units.
1
26. y = sin x
3
G. Stretch horizontally by a
factor of 3.
27. y = sin ( x + 3) + 2
H. Shift left 3 units, then
shift upward 2 units.
28. y = − sin ( x − 2 ) + 3
I.
Shift left 3 units.
29. y = − sin ( x ) − 3
J.
Shift downward 3 units.
30. y = sin ( x ) + 3
K. Stretch vertically by a
factor of 3.
L. Shrink vertically by a
1
factor of 3 .
69
Exercise Set 5.2: Graphs of the Sine and Cosine Functions
Matching. The left-hand column contains equations
that represent transformations of f ( x ) = cos ( x ) .
37. f ( x ) = 4 sin ( x )
Match the equations on the left with the description on
the right of how to obtain the graph of y = g ( x ) from
38. f ( x ) = 3cos ( x )
the graph of f .
31. y = cos 2 ( x − 8 )
32. y = cos ( 2 x − 8 )
33. y = cos ( 2 x ) − 8
34. y = cos 12 ( x − 8 )
35. y = cos 12 x − 8
36. y = cos ( x ) − 8
1
2
A. Stretch horizontally by a
factor of 2, then shift
right 16 units.
1
factor of 2 , then shift
right 4 units.
41. f ( x ) = 6 sin ( x ) − 2
42. f ( x ) = 4 cos ( x ) + 3
43. f ( x ) = cos ( 2 x )
C. Shrink horizontally by a
1
factor of 2 , then shift
downward 8 units.
E. Stretch horizontally by a
factor of 2, then shift
downward 8 units.
F. Shrink horizontally by a
1
factor of 2 , then shift
right 8 units.
For each of the following functions,
70
40. f ( x ) = −2sin ( x )
B. Shrink horizontally by a
D. Stretch horizontally by a
factor of 2, then shift
right 8 units.
(a)
(b)
(c)
(d)
(e)
39. f ( x ) = −5 cos ( x )
State the period.
State the amplitude.
State the phase shift.
State the vertical shift.
Use transformations to sketch the graph of
the function over one period. For consistency
in solutions, perform the appropriate
transformations on the function
g ( x ) = sin ( x ) or h ( x ) = cos ( x ) where
44. f ( x ) = − sin ( 3 x )
x
45. f ( x ) = − cos
4
46. f ( x ) =
2
x
cos
5
3
47. f ( x ) =
4
sin (π x )
3
πx
48. f ( x ) = − cos
+4
3
πx
49. f ( x ) = sin
−1
2
πx
50. f ( x ) = 4 cos
+2
3
π
51. f ( x ) = cos x −
2
52. f ( x ) = sin ( x + π )
53. f ( x ) = −3cos (π x + π )
0 ≤ x ≤ 2π . (Note: The resulting graph may
not fall within this same interval.) The x-axis
should be labeled to show the transformations
of the intercepts of g ( x ) = sin ( x ) or
54. f ( x ) = 5sin ( 4π x + π )
h ( x ) = cos ( x ) , as well as any x-values where
56. f ( x ) = cos ( 3 x − π )
a maximum or minimum occurs. The y-axis
should be labeled to reflect the maximum and
minimum values of the function.
57. f ( x ) = −7 cos ( 4 x − π )
55. f ( x ) = sin ( 2 x + 3π )
3π
1
58. f ( x ) = 5sin x +
2
4
University of Houston Department of Mathematics
Exercise Set 5.2: Graphs of the Sine and Cosine Functions
2π
59. f ( x ) = −3sin 2π x −
+5
3
70.
8 y
6
4
2
π
60. f ( x ) = 5cos x + π + 2
3
−π/2 −π/3 −π/6
x
π/6
−2
π/3
π/2
2π/3 5π/6
−4
x π
61. f ( x ) = 4 sin − − 2
2 2
−6
−8
π
62. f ( x ) = −3cos 2 x − + 4
2
71.
4
y
2
63. f ( x ) = 7 sin ( − x )
Omit part (c)
x
−5
64. f ( x ) = 10 sin ( − x )
Omit part (c)
65. f ( x ) = −2sin ( −2π x )
Omit part (c)
66. f ( x ) = −6sin ( −3 x ) − 4
Omit part (c)
−4
−3
−2
−1
1
2
3
4
5
6
−2
−4
3 y
72.
2
π
67. f ( x ) = 4 cos − x + 3 Omit part(c)
2
1
x
−2
68. f ( x ) = 5sin (π − x )
−1
1
2
3
−1
Omit part(c)
−2
−3
For each of the following graphs,
(a) Give an equation of the form
f ( x ) = A sin ( Bx − C ) + D which could be
73.
4 y
2
x
used to represent the graph. (Note: C or D
may be zero. Answers vary.)
−π/4
−6
−8
y
74.
12 y
10
4
8
2
6
x
−3π/4
−π/2
−π/4
π/4
π/2
3π/4
4
π
−2
−4
−6
3π/4
−4
used to represent the graph. (Note: C or D
may be zero. Answers vary.)
6
π/2
−2
(b) Give an equation of the form
f ( x ) = A cos ( Bx − C ) + D which could be
69.
π/4
2
−4
−3
−2
−1
−2
x
1
2
3
4
5
−4
MATH 1330 Precalculus
71
Exercise Set 5.2: Graphs of the Sine and Cosine Functions
75.
6
y
4
2
x
−4
−3
−2
−1
1
2
3
4
5
6
−2
76.
y
4
2
x
−5π/2 −2π −3π/2 −π
−π/2
π/2
π
3π/2
2π
−2
−4
−6
Answer the following.
77. The depth of the water at a certain point near the
shore varies with the ocean tide. Suppose that the
high tide occurs today at 2AM with a depth of 6
meters, and the low tide occurs at 8:30AM with a
depth of 2 meters.
(a) Write a trigonometric equation that models
the depth, D, of the water t hours after
midnight.
(b) Use a calculator to find the depth of the
water at 11:15AM. (Round to the nearest
hundredth.)
78. The depth of the water at a certain point near the
shore varies with the ocean tide. Suppose that the
high tide occurs today at 4AM with a depth of 9
meters, and the low tide occurs at 11:45AM with
a depth of 3 meters.
(a) Write a trigonometric equation that models
the depth, D, of the water t hours after
midnight.
(b) Use a calculator to find the depth of the
water at 2:30PM. (Round to the nearest
hundredth.)
72
University of Houston Department of Mathematics
Exercise Set 5.3: Graphs of the Tangent, Cotangent, Secant, and
Cosecant Functions
(f) Give an interval, if one exists, on which
f ( x ) = tan ( x ) is increasing. Then give an
Answer the following.
1.
(a) Use your calculator to complete the
following chart. Round to the nearest hundredth.
If a value is undefined, state “Undefined.” Notice
that the x-values on the chart increase by
π .
increments of 12
y = tan ( x )
x
x
y = tan ( x )
π
0
2
π
7π
12
12
π
interval , if one exists, on which
f ( x ) = tan ( x ) is decreasing.
2.
(a) Use your calculator to complete the
following chart. Round to the nearest
hundredth. If a value is undefined, state
“Undefined.” Notice that the x-values on the
π .
chart increase by increments of 12
6
π
5π
6
12
5π
12
11π
12
6
7π
12
π
2π
3
π
3π
4
4
π
π
(b) Plot the points from part (a) to discover the
graph of f ( x ) = tan ( x ) .
y = cot ( x )
2
π
3
π
x
π
0
3π
4
4
y = cot ( x )
x
2π
3
3
5π
6
5π
12
11π
12
π
4.0 y
3.0
2.0
1.0
x
π/12
π/6
π/4
(b) Plot the points from part (a) to discover the
graph of g ( x ) = cot ( x ) .
π/3 5π/12 π/2 7π/12 2π/3 3π/4 5π/6 11π/12 π
−1.0
4.0 y
−2.0
3.0
−3.0
2.0
−4.0
1.0
x
π/12
π/6
π/4
π/3 5π/12 π/2 7π/12 2π/3 3π/4 5π/6 11π/12 π
−1.0
(c) Use the graph from part (b) to sketch an
extended graph of f ( x ) = tan ( x ) , where
−2π ≤ x ≤ 2π . Be sure to show the
intercepts as well as any asymptotes.
(d) State the domain and range of
f ( x ) = tan ( x ) . Do not base this answer
simply on the limited domain from part (c),
but the entire graph of f ( x ) = tan ( x ) .
(e) State the period of f ( x ) = tan ( x ) .
MATH 1330 Precalculus
−2.0
−3.0
−4.0
(c) Use the graph from part (b) to sketch an
extended graph of g ( x ) = cot ( x ) , where
−2π ≤ x ≤ 2π . Be sure to show the
intercepts as well as any asymptotes.
Continued on the next page…
73
Exercise Set 5.3: Graphs of the Tangent, Cotangent, Secant, and
Cosecant Functions
(c) Use the graph from part (b) to sketch an
extended graph of f ( x ) = sec ( x ) , where
(d) State the domain and range of
g ( x ) = cot ( x ) . Do not base this answer
−2π ≤ x ≤ 2π . Be sure to show the
intercepts as well as any local maxima and
minima.
simply on the limited domain from part (c),
but the entire graph of g ( x ) = cot ( x ) .
(e) State the period of g ( x ) = cot ( x ) .
(d) On the graph from part (c), superimpose the
graph of h ( x ) = cos ( x ) lightly or in another
(f) Give an interval, if one exists, on which
g ( x ) = cot ( x ) is increasing. Then give an
color. What do you notice? How can this
serve as an aid in sketching the graph of
f ( x ) = sec ( x ) ?
interval , if one exists, on which
g ( x ) = cot ( x ) is decreasing.
3.
(e) State the domain and range of
f ( x ) = sec ( x ) . Base this answer on the
(a) Use your calculator to complete the
following chart. Round to the nearest
hundredth. If a value is undefined, state
“Undefined.”
y = sec ( x )
x
x
0
π
π
7π
6
6
π
entire graph of f ( x ) = sec ( x ) , not only on
the partial graphs obtained in parts (b)-(d).
y = sec ( x )
(f) State the period of f ( x ) = sec ( x ) .
(g) Give two intervals on which f ( x ) = sec ( x )
is increasing.
5π
4
4
π
4π
3
3
π
2
3π
2
2π
3
5π
3
3π
4
7π
4
5π
6
11π
6
4.
(a) Use your calculator to complete the
following chart. Round to the nearest
hundredth. If a value is undefined, state
“Undefined.”
x
π
π
7π
6
π
4
π
(b) Plot the points from part (a) to discover the
graph of f ( x ) = sec ( x ) .
1.2
1.0
3
π
y
0.8
0.6
0.4
0.2
−0.2
−0.4
−0.6
−0.8
x
π/6
π/3
π/2
2π/3 5π/6
π
7π/6 4π/3 3π/2 5π/3 11π/6 2π
x
0
6
2π
y = csc ( x )
y = csc ( x )
5π
4
4π
3
2
3π
2
2π
3
5π
3
3π
4
7π
4
5π
6
11π
6
2π
−1.0
−1.2
Continued on the next page…
74
University of Houston Department of Mathematics
Exercise Set 5.3: Graphs of the Tangent, Cotangent, Secant, and
Cosecant Functions
(b) Plot the points from part (a) to discover the
graph of g ( x ) = csc ( x ) .
1.2
1.0
Matching. The left-hand column contains equations
that represent transformations of f ( x ) = tan ( x ) .
Match the equations on the left with the description on
the right of how to obtain the graph of y = g ( x ) from
y
the graph of f .
0.8
0.6
0.4
0.2
−0.2
x
π/6
π/3
π/2
2π/3 5π/6
π
5.
y = tan ( x − 2 )
6.
y = tan ( x ) − 2
7.
y = tan ( 2 x )
8.
1
y = tan ( x )
2
9.
y = tan ( 2 x − 8 )
7π/6 4π/3 3π/2 5π/3 11π/6 2π
−0.4
−0.6
−0.8
−1.0
−1.2
A. Shrink horizontally by a
factor of 12 .
B. Reflect in the x-axis,
then shift upward 2
units.
C. Shift right 2 units.
(c) Use the graph from part (b) to sketch an
extended graph of g ( x ) = csc ( x ) , where
−2π ≤ x ≤ 2π . Be sure to show the
intercepts as well as any local maxima and
minima.
D. Reflect in the x-axis,
stretch vertically by a
factor of 2, shrink
horizontally by a factor
of 14 , then shift right 1
unit.
10. y = 2 tan ( x )
(d) On the graph from part (c), superimpose the
graph of h ( x ) = sin ( x ) lightly or in another
color. What do you notice? How can this
serve as an aid in sketching the graph of
g ( x ) = csc ( x ) ?
(e) State the domain and range of
g ( x ) = csc ( x ) . Base this answer on the
1
11. y = tan x
2
x
12. y = tan − 4
2
(f) State the period of g ( x ) = csc ( x ) .
F. Stretch horizontally by a
factor of 2.
G. Shrink horizontally by a
factor of 12 , then shift
13. y = −2 tan ( 4 x − 4 )
entire graph of g ( x ) = csc ( x ) , not only on
the partial graphs obtained in parts (b)-(d).
E. Stretch horizontally by a
factor of 2, then shift
downward 4 units.
right 4 units.
H. Shift downward 2 units.
14. y = − tan ( x ) + 2
I.
Stretch vertically by a
factor of 2.
J.
Shrink vertically by
factor of 12 .
(g) Give two intervals on which g ( x ) = csc ( x )
is increasing.
Continued on the next page…
MATH 1330 Precalculus
75
Exercise Set 5.3: Graphs of the Tangent, Cotangent, Secant, and
Cosecant Functions
Matching. The left-hand column contains equations
that represent transformations of f ( x ) = csc ( x ) .
appropriate transformations on g ( x ) = tan ( x )
or h ( x ) = cot ( x ) within the following
Match the equations on the left with the description on
the right of how to obtain the graph of y = g ( x ) from
intervals. (Note: The resulting graph may not
fall within these intervals.)
the graph of f .
1
15. y = csc ( 3 x + π )
2
16. y =
1
csc 3 ( x + π )
2
17. y = 2 csc ( 3 x ) + π
1
π
18. y = 2 csc x +
3
3
π
1
19. y = − csc x +
3
3
1
20. y = − csc x + π
3
g ( x ) = tan ( x ) , where −
A. Reflect in the x-axis,
stretch horizontally by a
factor of 3, then shift
left π units.
B. Shrink vertically by a
factor of 12 , shrink
π
2
< x<
π
2
h ( x ) = cot ( x ) , where 0 < x < π
21. f ( x ) = 5 tan ( x )
22. f ( x ) = 4 cot ( x )
horizontally by a factor
of
1
3
, then shift left
π
3
units.
C. Stretch vertically by a
factor of 2, shrink
horizontally by a factor
of 13 , then shift upward
π units.
D. Stretch vertically by a
factor of 2, stretch
horizontally by a factor
of 3, then shift left
π
3
units.
23. f ( x ) = 4 tan ( x ) + 1
24. f ( x ) = 5cot ( x ) − 4
25. f ( x ) = 2 cot ( 3x )
26. f ( x ) = −4 tan ( 2 x )
x
27. f ( x ) = 5cot − + 2
4
x
28. f ( x ) = −3cot − 5
2
29. f ( x ) = −7 tan ( 4π x ) − 3
E. Reflect in the x-axis,
stretch horizontally by a
factor of 3, then shift
upward π units.
F. Shrink vertically by a
factor of 12 , shrink
horizontally by a factor
of 13 , then shift left π
units.
30. f ( x ) = 2 tan ( −π x ) + 1
π
31. f ( x ) = cot x +
3
π
32. f ( x ) = − tan x −
2
x π
33. f ( x ) = 2 tan − − 5
2 4
For each of the following functions,
(a) State the period.
(b) Use transformations to sketch the graph of
the function over one period. Label any
asymptotes clearly. Be sure to show the exact
transformation of each point on the basic
graphs of g ( x ) = tan ( x ) or h ( x ) = cot ( x )
whose x-value is a multiple of
π
4
. For
π
1
34. f ( x ) = 5 tan x + − 2
4
4
π
35. f ( x ) = −3cot π x + + 2
2
π
36. f ( x ) = 4 cot π x − + 3
3
consistency in solutions, perform the
76
University of Houston Department of Mathematics
Exercise Set 5.3: Graphs of the Tangent, Cotangent, Secant, and
Cosecant Functions
Answer the following.
37. The graph of f ( x ) = sin ( x ) can be useful in
sketching transformations of the graph of
g ( x ) = csc ( x ) . Answer the following, using the
interval −2π ≤ x ≤ 2π for each graph.
(a) Sketch the graphs of f ( x ) = sin ( x ) and
g ( x ) = csc ( x ) on the same set of axes.
(b) Where are the x-intercepts of
f ( x ) = sin ( x ) ?
(c) Where are the asymptotes of
g ( x ) = csc ( x ) ? Explain the relationship
between the answers in parts (b) and (c).
(d) On a different set of axes, sketch the graph
of h ( x ) = −2 sin ( 3x ) .
(e) Use the graph in part (d) to sketch the graph
of p ( x ) = −2 csc ( 3x ) on the same set of
axes.
(f) On a different set of axes, sketch the graph
π
of q ( x ) = 4 sin x − − 3 .
2
(g) Use the graph in part (f) to sketch the graph
π
of r ( x ) = 4 csc x − − 3 on the same set
2
of axes.
(e) Use the graph in part (d) to sketch the graph
of p ( x ) = 3sec ( 2 x ) + 1 on the same set of
axes.
(f) On a different set of axes, sketch the graph
of q ( x ) = −5cos ( 2 x + π ) .
(g) Use the graph in part (f) to sketch the graph
of r ( x ) = −5sec ( 2 x + π ) on the same set of
axes.
For each of the following functions,
(a) State the period.
(b) Use transformations to sketch the graph of
the function over one period. The x-axis
should be labeled to reflect the location of any
x-values where a relative maximum or
minimum occurs. The y-axis should be labeled
to reflect any maximum and minimum values
of the function. Label any asymptotes clearly.
For consistency in solutions, perform the
appropriate transformations on g ( x ) = sec ( x )
where 0 ≤ x ≤ 2π , or h ( x ) = csc ( x ) , where
0 < x < 2π . (Note: The resulting graph may
not fall within these intervals.)
39. f ( x ) = 8sec ( x )
40. f ( x ) = 5 csc ( x )
41. f ( x ) = −4 csc ( x ) + 5
38. The graph of f ( x ) = cos ( x ) can be useful in
sketching transformations of the graph of
g ( x ) = sec ( x ) . Answer the following, using the
interval −2π ≤ x ≤ 2π for each graph.
(a) Sketch the graphs of f ( x ) = cos ( x ) and
g ( x ) = sec ( x ) on the same set of axes.
(b) Where are the x-intercepts of
f ( x ) = cos ( x ) ?
(c) Where are the asymptotes of
g ( x ) = sec ( x ) ? Explain the relationship
between the answers in parts (b) and (c).
(d) On a different set of axes, sketch the graph
of h ( x ) = 3cos ( 2 x ) + 1 .
MATH 1330 Precalculus
42. f ( x ) = 2 sec ( x ) − 3
x
43. f ( x ) = − sec
2
2
x
44. f ( x ) = − sec
3
5
45. f ( x ) = 7 sec ( 3x ) − 1
46. f ( x ) = −4 csc ( 2 x ) + 3
47. f ( x ) =
7
csc ( 2π x ) + 1
2
48. f ( x ) = −2 csc ( −π x ) − 3
77
Exercise Set 5.3: Graphs of the Tangent, Cotangent, Secant, and
Cosecant Functions
πx
49. f ( x ) = 4 csc −
+5
3
60.
y
2
x
−π
π x
50. f ( x ) = 2 sec
−6
4
−3π/4 −π/2 −π/4
π/4
π/2
3π/4
π
−2
−4
−6
51. f ( x ) = −4sec ( x − π ) − 2
−8
−10
π
52. f ( x ) = 8 csc x + + 3
2
61.
π
53. f ( x ) = −2 csc π x +
2
2
y
x
−5π/8 −π/2 −3π/8 −π/4 −π/8
π/8
π/4 3π/8 π/2
−2
−4
54. f ( x ) = 4 csc ( 2π x − π )
−6
−8
55. f ( x ) = 3sec (π − 3 x )
−10
−12
56. f ( x ) = −9sec ( 3π − 2 x )
x π
57. f ( x ) = 2 csc − + 4
3 2
π x 2π
58. f ( x ) = −9sec
+
3
6
y
62.
8
6
4
+2
2
x
−π/3
−π/4
−π/6
−π/12
π/12
π/6
−2
−4
For each of the following graphs,
(a) Give an equation of the form
f ( x ) = A tan ( Bx − C ) + D which could be
For each of the following graphs,
used to represent the graph. (Note: C or D
may be zero. Answers vary.)
(a) Give an equation of the form
f ( x ) = A sec ( Bx − C ) + D which could be
(b) Give an equation of the form
f ( x ) = A cot ( Bx − C ) + D which could be
used to represent the graph. (Note: C or D
may be zero. Answers vary.)
used to represent the graph. (Note: C or D
may be zero. Answers vary.)
59.
(b) Give an equation of the form
f ( x ) = A csc ( Bx − C ) + D which could be
y
f
used to represent the graph. (Note: C or D
may be zero. Answers vary.)
8
6
4
63.
6 y
2
4
x
−π
−3π/4 −π/2 −π/4
π/4
π/2
3π/4
2
π
x
−2
−4
−3π/2
−π
−π/2
π/2
π
3π/2
2π
−2
−4
−6
−8
78
University of Houston Department of Mathematics
Exercise Set 5.3: Graphs of the Tangent, Cotangent, Secant, and
Cosecant Functions
64.
10 y
8
6
4
2
x
−π
−π/2
π/2
π
−2
−4
65.
y
10
8
6
4
2
x
−5
−4
−3
−2
−1
1
2
3
−2
−4
66.
y
2
x
−2.0 −1.5 −1.0 −0.5
0.5
1.0 1.5 2.0 2.5 3.0
−2
−4
−6
−8
−10
−12
MATH 1330 Precalculus
79
Exercise Set 5.4: Inverse Trigonometric Functions
Exercises 1-6 help to establish the graphs of the
inverse trigonometric functions, and part (h) of each
exercise shows a shortcut for remembering the range.
Answer the following.
1.
(d) Plot the points from part (c) on the axes
below. Then use those points along with the
graph from part (b) to sketch the graph of
x = cos ( y ) .
(a) Complete the following chart. Round to the
nearest hundredth.
2π
y
3π/2
x
y = cos ( x )
y = cos ( x )
x
π
0
π/2
π
−
2
π
π
x
2
−1.2 −1.0 −0.8 −0.6 −0.4 −0.2
−
2π
−2π
0.4
0.6
0.8
1.0
1.2
−π/2
−π
3π
2
0.2
−π
3π
2
−3π/2
−2π
(b) Plot the points from part (a) on the axes
below. Then use those points along with
previous knowledge of trigonometric graphs
to sketch the graph of y = cos ( x ) .
1.2
0.8
0.6
0.4
0.2
−3π/2
−π
−π/2
−0.2
−0.4
(f) Using the graph of x = cos ( y ) from part
(d), sketch the portion of the graph with
each of the following restricted ranges. Then
state if each graph represents a function.
y
1.0
−2π
(e) Is the inverse relation in part (d) a function?
Why or why not?
[ 0, π ]
ii.
iii.
[ −π , 0]
π π
iv. − ,
2 2
x
π/2
π
3π/2
2π
−0.6
−0.8
−1.0
(g) One of the functions from part (f) is the
graph of f ( x ) = cos −1 ( x ) . Which one?
−1.2
(c) Use the chart from part (a) to complete the
following chart for x = cos ( y ) .
x = cos ( y )
y
π 3π
2, 2
i.
x = cos ( y )
y
0
π
2
π
−
π
(h) Below is a way to remember the range of
f ( x ) = cos −1 ( x ) without drawing a graph:
i.
2
−π
3π
2
−
2π
−2π
Continued in the next column…
Explain why that particular range may be
the most reasonable choice.
3π
2
ii.
Draw a unit circle on the coordinate
plane. In each quadrant, write “ + ” if
the cosine of angles in that quadrant are
positive, and write “ − ” if the cosine of
angles in that quadrant are negative.
Using the diagram from part i, find two
consecutive quadrants where one is
labeled “ + ” and the other is labeled
“ − ”. Write down any other pairs of
consecutive quadrants where this
occurs.
Continued on the next page…
80
University of Houston Department of Mathematics
Exercise Set 5.4: Inverse Trigonometric Functions
iii. Of the answers from part ii, choose the
pair of quadrants that includes Quadrant
I; this illustrates the range of
f ( x ) = cos −1 ( x ) . Write the range in
(d) Plot the points from part (c) on the axes
below. Then use those points along with the
graph from part (b) to sketch the graph of
x = sin ( y ) .
interval notation.
2.
2π
3π/2
(a) Complete the following chart. Round to the
nearest hundredth.
x
y = sin ( x )
π
y = sin ( x )
x
y
π/2
x
0
−1.2 −1.0 −0.8 −0.6 −0.4 −0.2
π
−
2
2
−π
3π
2
3π
−
2
2π
−2π
0.4
0.6
0.8
1.0
1.2
−π/2
π
π
0.2
−π
−3π/2
−2π
(e) Is the inverse relation in part (d) a function?
Why or why not?
(b) Plot the points from part (a) on the axes
below. Then use those points along with
previous knowledge of trigonometric graphs
to sketch the graph of y = sin ( x ) .
1.2
(f) Using the graph of x = sin ( y ) from part (d),
sketch the portion of the graph with each of
the following restricted ranges. Then state if
each graph represents a function.
y
1.0
0.8
0.6
[ 0, π ]
ii.
iii.
[ −π , 0]
π π
iv. − ,
2 2
0.4
0.2
−2π
−3π/2
−π
−π/2
−0.2
−0.4
x
π/2
π
3π/2
2π
−0.6
(g) One of the functions from part (f) is the
graph of f ( x ) = sin −1 ( x ) . Which one?
−0.8
−1.0
−1.2
(c) Use the chart from part (a) to complete the
following chart for x = sin ( y ) .
x = sin ( y )
y
π 3π
2, 2
i.
x = sin ( y )
y
2
π
(h) Below is a way to remember the range of
f ( x ) = sin −1 ( x ) without drawing a graph:
i.
0
π
Explain why that particular range may be
the most reasonable choice.
−
π
2
−π
ii.
3π
2
−
2π
−2π
Continued in the next column…
3π
2
Draw a unit circle on the coordinate
plane. In each quadrant, write “ + ” if
the sine of angles in that quadrant are
positive, and write “ − ” if the sine of
angles in that quadrant are negative.
Using the diagram from part i, find two
consecutive quadrants where one is
labeled “ + ” and the other is labeled
“ − ”. Write down any other pairs of
consecutive quadrants where this
occurs.
Continued on the next page…
MATH 1330 Precalculus
81
Exercise Set 5.4: Inverse Trigonometric Functions
iii. Of the answers from part ii, choose the
pair of quadrants that includes Quadrant
I; this illustrates the range of
f ( x ) = sin −1 ( x ) . Write the range in
(d) Plot the points from part (c) on the axes
below. Then use those points along with the
graph from part (b) to sketch the graph of
x = tan ( y ) .
interval notation.
π
y
3π/4
3.
(a) Complete the following chart. Round to the
nearest hundredth. If a value is undefined,
state “Undefined.”
x
y = tan ( x )
y = tan ( x )
x
π/2
π/4
x
−1.2 −1.0 −0.8 −0.6 −0.4 −0.2
0
0.4
0.6
0.8
1.0
1.2
−π/4
π
−
4
π
−
2
π
−π/2
4
−3π/4
π
2
3π
4
3π
−
4
π
−π
−π
(e) Is the inverse relation in part (d) a function?
Why or why not?
(b) Plot the points from part (a) on the axes
below. Then use those points along with
previous knowledge of trigonometric graphs
to sketch the graph of y = tan ( x ) .
1.2
0.8
0.6
0.4
0.2
−3π/4
−π/2
−π/4
−0.2
−0.4
x
π/4
π/2
3π/4
π
−0.6
−0.8
−1.0
−1.2
(c) Use the chart from part (a) to complete the
following chart for x = tan ( y ) .
x = tan ( y )
y
(f) Using the graph of x = tan ( y ) from part
(d), sketch the portion of the graph with
each of the following restricted ranges. Then
state if each graph represents a function.
π π
0, ∪ , π
2 2
π π
ii. −π , − ∪ − , 0
2 2
π π
iii. 0, ∪ , π
2 2
π π
iv. − ,
2 2
i.
y
1.0
−π
0.2
x = tan ( y )
y
0
π
4
π
2
−
−
π
4
π
2
3π
4
3π
−
4
π
−π
Continued in the next column…
(g) One of the functions from part (f) is the
graph of f ( x ) = tan −1 x . Which one?
Explain why that particular range may be
the most reasonable choice.
(h) Below is a way to remember the range of
f ( x ) = tan −1 ( x ) without drawing a graph:
i.
Draw a unit circle on the coordinate
plane. In each quadrant, write “ + ” if
the tangent of angles in that quadrant
are positive, and write “ − ” if the
tangent of angles in that quadrant are
negative. Also make a note next to any
quadrantal angles for which the tangent
function is undefined.
Continued on the next page…
82
University of Houston Department of Mathematics
Exercise Set 5.4: Inverse Trigonometric Functions
ii.
Using the diagram from part i, find two
consecutive quadrants where one is
labeled “ + ” and the other is labeled
“ − ”. Write down any other pairs of
consecutive quadrants where this
occurs.
iii. Of the answers from part ii, write down
any pairs of quadrants that include
Quadrant I. (There are two pair.)
Suppose for a moment that either pair
could illustrate the range of
f ( x ) = tan −1 ( x ) , and write down the
ranges for each in interval notation,
remembering to consider where the
function is undefined. Then choose the
range that is represented by a single
interval. This is the range of
f ( x ) = tan −1 ( x ) .
(c) Use the chart from part (a) to complete the
following chart for x = cot ( y ) .
x = cot ( y )
x = cot ( y )
y
0
π
−
4
π
−
2
y = cot ( x )
2
3π
−
4
π
−π
y
3π/4
π/2
y = cot ( x )
x
4
π
(d) Plot the points from part (c) on the axes
below. Then use those points along with the
graph from part (b) to sketch the graph of
x = cot ( y ) .
(a) Complete the following chart. Round to the
nearest hundredth. If a value is undefined,
state “Undefined.”
x
π
3π
4
π
4.
y
π/4
x
0
−1.2 −1.0 −0.8 −0.6 −0.4 −0.2
π
−
4
π
0.2
0.4
0.6
0.8
1.0
1.2
−π/4
4
−π/2
π
−
2
π
2
3π
4
3π
−
4
π
−π
−3π/4
−π
(e) Is the inverse relation in part (d) a function?
Why or why not?
(b) Plot the points from part (a) on the axes
below. Then use those points along with
previous knowledge of trigonometric graphs
to sketch the graph of y = cot ( x ) .
1.2
(f) Using the graph of x = cot ( y ) from part
(d), sketch the portion of the graph with
each of the following restricted ranges. Then
state if each graph represents a function.
y
1.0
( 0, π )
ii.
iii.
( −π , 0 )
π π
iv. − , 0 ∪ 0,
2 2
0.8
0.6
0.4
0.2
−π
−3π/4
−π/2
−π/4
−0.2
−0.4
x
π/4
−0.6
−0.8
−1.0
−1.2
Continued in the next column…
MATH 1330 Precalculus
π/2
3π/4
π π
− , 0 ∪ 0,
2 2
i.
π
(g) One of the functions from part (f) is the
graph of f ( x ) = cot −1 x . Which one?
Explain why that particular range may be
the most reasonable choice.
Continued on the next page…
83
Exercise Set 5.4: Inverse Trigonometric Functions
(h) Below is a way to remember the range of
f ( x ) = cot −1 ( x ) without drawing a graph:
i.
Draw a simple coordinate plane by
sketching the x and y-axes. In each
quadrant, write “ + ” if the cotangent of
numbers in that quadrant are positive,
and write “ − ” if the cotangent of
numbers in that quadrant are negative.
Also make a note next to any quadrantal
angles for which the cotangent function
is undefined.
ii. Using the diagram from part i, find two
consecutive quadrants where one is
labeled “ + ” and the other is labeled
“ − ”. Write down any other pairs of
consecutive quadrants where this
occurs.
iii. Of the answers from part ii, write down
any pairs of quadrants that include
Quadrant I. (There are two pair.)
Suppose for a moment that either pair
could illustrate the range of
f ( x ) = cot −1 ( x ) , and write down the
ranges for each in interval notation,
remembering to consider where the
function is undefined. Then choose the
range that is represented by a single
interval. This is the range of
f ( x ) = cot −1 ( x ) .
5.
(a) Complete the following chart. Round to the
nearest hundredth. If a value is undefined,
state “Undefined.”
x
y = csc ( x )
x
(b) Plot the points from part (a) on the axes
below. Then use those points along with
previous knowledge of trigonometric graphs
to sketch the graph of y = csc ( x ) .
4
y
3
2
1
x
−2π
−3π/2
−π
−π/2
π/2
π
3π/2
2π
−1
−2
−3
−4
(c) Use the chart from part (a) to complete the
following chart for x = csc ( y ) .
x = csc ( y )
x = csc ( y )
y
y
0
π
−
2
π
π
2
−π
3π
2
−
3π
2
2π
−2π
(d) Plot the points from part (c) on the axes
below. Then use those points along with the
graph from part (b) to sketch the graph of
x = csc ( y ) .
2π
y = csc ( x )
y
3π/2
0
π
π
2
π
−
π
2
π/2
x
−π
−1.2 −1.0 −0.8 −0.6 −0.4 −0.2
3π
2
−
3π
2
2π
−2π
0.2
0.4
0.6
0.8
1.0
1.2
−π/2
−π
−3π/2
−2π
Continued in the next column…
(e) Is the inverse relation in part (d) a function?
Why or why not?
Continued on the next page…
84
University of Houston Department of Mathematics
Exercise Set 5.4: Inverse Trigonometric Functions
(f) Using the graph of x = csc ( y ) from part
(d), sketch the portion of the graph with
each of the following restricted ranges. Then
state if each graph represents a function.
i.
iii.
(b) Plot the points from part (a) on the axes
below. Then use those points along with
previous knowledge of trigonometric graphs
to sketch the graph of y = sec ( x ) .
( 0, π )
π
3π
2 , π ∪ π , 2
4
ii.
( −π , 0 )
π π
iv. − , 0 ∪ 0,
2 2
1
2
x
−2π
−3π/2
−π
−π/2
π/2
(h) Below is a way to remember the range of
f ( x ) = csc−1 ( x ) without drawing a graph:
i.
Draw a unit circle on the coordinate
plane. In each quadrant, write “ + ” if
the cosecant of angles in that quadrant
are positive, and write “ − ” if the
cosecant of angles in that quadrant are
negative. Also make a note next to any
quadrantal angles for which the
cosecant function is undefined.
ii. Using the diagram from part i, find two
consecutive quadrants where one is
labeled “ + ” and the other is labeled
“ − ”. Write down any other pairs of
consecutive quadrants where this
occurs.
iii. Of the answers from part ii, choose the
pair of quadrants that includes Quadrant
I; this illustrates the range of
f ( x ) = csc−1 ( x ) . Consider where the
π
3π/2
2π
−1
−2
(g) One of the functions from part (f) is the
graph of f ( x ) = csc−1 ( x ) . Which one?
Explain why that particular range may be
the most reasonable choice.
y
3
−3
−4
(c) Use the chart from part (a) to complete the
following chart for x = sec ( y ) .
x = sec ( y )
x = sec ( y )
y
y
0
π
−
2
π
π
2
−π
3π
2
−
3π
2
2π
−2π
(d) Plot the points from part (c) on the axes
below. Then use those points along with the
graph from part (b) to sketch the graph of
x = sec ( y ) .
2π
y
3π/2
function is undefined, and then write the
range in interval notation.
π
π/2
6.
(a) Complete the following chart.
x
y = sec ( x )
x
x
y = sec ( x )
−1.2 −1.0 −0.8 −0.6 −0.4 −0.2
0.2
0.4
0.6
0.8
1.0
1.2
−π/2
0
π
2
π
−π
−
π
2
−3π/2
−π
−2π
3π
2
−
3π
2
2π
−2π
Continued in the next column…
MATH 1330 Precalculus
(e) Is the inverse relation in part (d) a function?
Why or why not?
Continued on the next page…
85
Exercise Set 5.4: Inverse Trigonometric Functions
(f) Using the graph of x = sec ( y ) from part
(d), sketch the portion of the graph with
each of the following restricted ranges. Then
state if each graph represents a function.
i.
ii.
π π
0, 2 ∪ 2 , π
π 3π
,
2 2
π π
iii. −π , − ∪ − , 0
2 2
π π
iv. − ,
2 2
(g) One of the functions from part (f) is the
graph of f ( x ) = sec−1 ( x ) . Which one?
Explain why that particular range may be
the most reasonable choice.
(h) Below is a way to remember the range of
f ( x ) = sec−1 ( x ) without drawing a graph:
i.
Draw a unit circle on the coordinate
plane. In each quadrant, write “ + ” if
the secant of angles in that quadrant are
positive, and write “ − ” if the secant of
angles in that quadrant are negative.
Also make a note next to any quadrantal
angles for which the secant function is
undefined.
ii. Using the diagram from part i, find two
consecutive quadrants where one is
labeled “ + ” and the other is labeled
“ − ”. Write down any other pairs of
consecutive quadrants where this
occurs.
iii. Of the answers from part ii, choose the
pair of quadrants that includes Quadrant
I; this illustrates the range of
f ( x ) = sec−1 ( x ) . Consider where the
function is undefined, and then write the
range in interval notation.
Inverse functions may be easier to remember if they
are translated into words. For example:
sin −1 ( x ) = the number (in the interval − π2 , π2 )
whose sine is x.
Translate each of the following expressions into words
(and include the range of the inverse function in
parentheses as above). Then find the exact value of
each expression. Do not use a calculator.
7.
1
(a) cos −1 −
2
(b) tan −1 (1)
8.
2
(a) sin −1
2
(b) cot −1 ( −1)
9.
(a) sin −1 ( −1)
(b) sec−1
3
10. (a) cos −1 −
2
(b) csc −1 ( −2 )
11. sin sin −1 ( 0.2 )
12. cos arccos ( 0.7 )
13. tan arctan ( 4 )
14. csc csc−1 ( 3.5 )
Find the exact value of each of the following
expressions. Do not use a calculator. If undefined,
state, “Undefined.”
1
15. (a) sin −1
2
(b) cos −1 (1)
3
16. (a) cos −1
2
(b) arcsin (1)
17. (a) arccos
( 3)
(b) sin −1 ( 0 )
(
)
18. (a) sin −1 − 2
86
( 2)
(b) cos −1 ( −1)
( 3)
19. (a) tan −1 ( −1)
(b) tan −1
20. (a) arctan ( 0 )
3
(b) tan −1 −
3
University of Houston Department of Mathematics
Exercise Set 5.4: Inverse Trigonometric Functions
3
21. (a) cot −1 −
3
(b) cot −1 ( 0 )
22. (a) cot −1 (1)
(b) cot −1 − 3
23. (a) sec−1 ( −2 )
(b) arccsc ( −1)
(
−1
Use a calculator to find the value of each of the
following expressions. Round each answer to the
nearest thousandth. If undefined, state, “Undefined.”
)
2 3
24. (a) csc
3
(b) sec
2
25. (a) csc −1
2
2 3
(b) sec−1 −
3
26. (a) sec
−1
−1
( 0)
27. (a) sec
−1
1
(b) sec −
2
(− 2 )
( x) =
cot −1 ( −2.1) ≈ π − 0.444 ≈ 2.698 . A more exact answer can be
obtained by rounding later in the calculation:
1
π − tan −1 ≈ 2.697 . (Either answer is acceptable.)
2.1
(b) cos −1 ( −0.24 )
4
33. (a) sin −1 −
7
7
(b) arccos −
9
34. (a) cos −1 ( −3)
1
(b) sin −1 −
5
35. (a) tan −1 ( 4 )
(b) arctan ( −0.12 )
36. (a) arctan ( 0.28 )
(b) tan −1 ( −30 )
−1
3
37. (a) cot −1
8
(b) cot −1 ( −0.8)
1
38. (a) cot −1 ( 6 )
11
(b) cot −1 −
5
1
39. (a) cot −1 −
5
(b) arccot ( −10 )
40. (a) cot −1 ( −7 )
(b) cot −1 ( −.62 )
41. (a) csc −1 ( 2.4 )
10
(b) sec−1 −
3
cos −1 ( x )
−1
1
sin
−1
( x)
1
29. (a) cot ( x ) = tan
x
(b) cot −1 ( x ) =
π
number, the solution lies in the interval , π . Therefore,
2
32. (a) sin −1 ( 0.9 )
1
28. (a) csc −1 ( x ) = sin −1
x
−1
−2.1 .” Since the range of the inverse cotangent function is
−2.1 , a negative
(b) sin −1 ( −0.8 )
1
(b) sec ( x ) = cos
x
(b) csc −1 ( x ) =
cot −1 ( −2.1) can be translated as “the cotangent of a number is
31. (a) cos −1 ( 0.37 )
1
−1
1
reference angle) by finding tan −1
≈ 0.444 . The expression
2.1
( 0, π ) , and the cotangent of the number is
Answer True or False. Assume that all x-values are in
the domain of the given inverse functions.
−1
Note: To find the inverse cotangent of a negative number, such
as cot −1 ( −2.1) , first find a reference number (to be likened to a
tan −1 ( x )
1
30. Explain why cot −1 ( x ) ≠ tan −1 . Are there
x
1
any values for which cot −1 ( x ) = tan −1 ? If
x
so, which values?
MATH 1330 Precalculus
87
Exercise Set 5.4: Inverse Trigonometric Functions
3
42. (a) sec−1
4
(b) csc −1 ( −7.2 )
43. (a) sec−1 ( −0.71)
31
(b) csc −1 −
5
44. (a) csc −1 ( −9.5 )
(b) arcsec ( 8.8)
3
56. (a) tan cos −1 −
2
(b) cos tan −1 − 3
(
3
57. (a) csc tan −1 −
3
Find the exact value of each of the following
expressions. Do not use a calculator. If undefined,
state, “Undefined.”
45. (a) sin sin −1 ( −0.84 )
)
(b) cos cos −1
( 3 )
(b) tan sec−1 ( −2 )
1
58. (a) cot cos −1 −
2
(b) sin csc−1 − 2
(
46. (a) sin sin −1 ( −5)
2
(b) cos cos −1
3
47. (a) csc csc−1 ( 3.5 )
(b) cot cot −1 ( −7 )
48. (a) sec sec
(b) tan tan
−1
( −0.3)
−1
( 0.5)
3
49. (a) sin cos −1
5
8
(b) tan sin −1
17
12
50. (a) cos tan −1
5
7
(b) cot cos −1
25
2
51. (a) sec tan −1
7
(b) cot csc−1 ( 5 )
52. (a) csc cot −1 ( 4 )
7
(b) tan sec −1
4
)
4π
59. (a) cos −1 cos
3
π
60. (a) cos −1 cos
6
2π
(b) sin −1 sin
3
7π
61. (a) sec−1 sec
6
3π
(b) tan −1 tan
4
7π
62. (a) cot −1 cot
4
4π
(b) csc −1 csc
3
Sketch the graph of each function.
1
(b) tan cos −1 −
6
1
54. (a) cos tan −1 −
7
3
(b) cot sin −1 −
7
64. y = arctan ( x + 1)
65. y = cot −1 ( x ) −
π
2
66. y = csc−1 ( x ) − π
67. y = arcsin ( x − 1) + π
68. y = cos −1 ( x + 2 ) +
88
π
(b) sin −1 sin
4
63. y = cos −1 ( x − 2 )
12
53. (a) sin cos −1 −
13
55. (a) sin tan −1 ( −1)
2
(b) cos sin −1 −
2
π
2
University of Houston Department of Mathematics
Exercise Set 6.1: Sum and Difference Formulas
Simplify each of the following expressions.
1.
sin (π − x )
2.
3π
cos x +
2
3.
π
cos x −
2
17. Given that tan (α ) = −3 and tan ( β ) =
2
,
5
evaluate tan (α + β ) .
18. Given that tan (α ) = −
4
1
and tan ( β ) = − ,
2
3
evaluate tan (α − β ) .
Simplify each of the following expressions as much as
possible without a calculator.
4.
sin (π + x )
5.
sin 60 − θ + sin 60 + θ
6.
cos 60 − θ + cos 60 + θ
7.
π
π
cos x − + cos x +
4
4
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
(
)
(
)
19. cos 55 cos 10 + sin 55 sin 10
(
)
(
)
20. cos 75 cos 15 − sin 75 sin 15
21. sin 45 cos 15 − cos 45 sin 15
22. sin 53 cos 7 + cos 53 sin 7
8.
9.
π
π
sin x + + sin x −
6
6
(
)
(
)
(
sin θ − 180 + sin θ + 180
(
10. cos 90 + θ + cos 90 − θ
π
π
π π
23. cos cos − sin sin
10
9
10 9
)
π
π
π π
24. sin cos − cos sin
5
7
5 7
)
13π
π
13π π
25. sin
cos + cos
sin
12
12
12 12
Answer the following.
3π
11. Given that tan (α ) = −4 , evaluate tan α +
.
4
π
12. Given that tan ( β ) = 2 , evaluate tan − β .
4
13. Given that tan ( x ) =
2
5π
, evaluate tan x −
3
4
28. cos ( 3α ) cos (α ) + sin ( 3α ) cos (α )
( ) ( )
1 − tan ( 32 ) tan ( 2 )
tan 32 + tan 2
.
15. Given that tan ( x ) = 5 and tan ( y ) = 6 , evaluate
tan ( x + y ) .
5π 17π
+ sin
sin
12 12
27. sin ( 2 A ) cos ( A ) − cos ( 2 A ) sin ( A )
.
1
7π
14. Given that tan ( x ) = − , evaluate tan x +
5
4
5π
17π
26. cos
cos
12
12
29.
( ) ( )
1 + tan ( 49 ) tan ( 4 )
tan 49 − tan 4
30.
16. Given that tan ( x ) = 4 and tan ( y ) = −2 ,
evaluate tan ( x − y ) .
MATH 1330 Precalculus
89
Exercise Set 6.1: Sum and Difference Formulas
38. For fractions with larger magnitude than those in
Exercises 36 and 37, it can be helpful to use
5π
π
tan
− tan
12
12
31.
5π
π
1 + tan
tan
12
12
larger multiples of
34.
tan ( a − b ) + tan ( b )
(a)
5π
4
(b)
7π
4
(c)
9π
4
(d)
11π
4
contains a multiple of
tan ( 2c + 3d ) − tan ( d − c )
1 + tan ( 2c + 3d ) tan ( d − c )
Answer the following.
(b)
π
(c)
4
3π
(e)
4
6
2π
(d)
3
(f)
π
3
5π
6
of
π
4
5π
12
(b)
7π
12
(c)
13π
12
(d)
11π
12
37. Use the answers from Exercise 35 to write each
fraction below as the difference of two special
angles. (Hint: Each of the solutions contains a
multiple of
5π
12
(c) −
90
.)
(b)
25π
12
(c)
35π
12
(d)
43π
12
40. Use the answers from Exercises 35 and 38 to
write each fraction below as the difference of
two special angles. (Hint: Each of the solutions
contains a multiple of
19π
12
23π
(c) −
12
π
4
.)
25π
12
29π
(d) −
12
(b)
.)
(a)
(a)
4
19π
12
(a)
36. Use the answers from Exercise 35 to write each
fraction below as the sum of two special angles.
(Hint: Each of the solutions contains a multiple
π
(a)
35. Rewrite each special angle below so that it has a
denominator of 12.
π
. Rewrite each special
39. Use the answers from Exercises 35 and 38 to
write each fraction below as the sum of two
special angles. (Hint: Each of the solutions
1 − tan ( a − b ) tan ( b )
(a)
4
angle below so that it has a denominator of 12.
13π
7π
tan
+ tan
12
12
32.
13π
7π
1 − tan
tan
12
12
33.
π
7π
12
π
4
.)
(b)
7π
12
(d) −
5π
12
41. Use the answers from Exercises 35 and 38, along
with their negatives to write each fraction below
as the difference, x − y , of two special angles,
where x is negative and y is positive.
(a) −
7π
12
(b) −
13π
12
(c) −
29π
12
(d) −
41π
12
42. Use the answers from Exercises 35 and 38, along
with their negatives to write each fraction below
as the difference, x − y , of two special angles,
where x is negative and y is positive.
(a) −
5π
12
(b) −
19π
12
(c) −
31π
12
(d) −
43π
12
University of Houston Department of Mathematics
Exercise Set 6.1: Sum and Difference Formulas
Answer the following.
43. Use a sum or difference formula to prove that
sin ( −θ ) = − sin (θ ) .
44. Use a sum or difference formula to prove that
cos ( −θ ) = cos (θ ) .
45. Use a sum or difference formula to prove that
tan ( −θ ) = − tan (θ ) .
46. Use a sum or difference formula to prove that
π
cos − θ = sin (θ ) .
2
37π
61. tan
12
29π
62. tan −
12
Answer the following. (Hint: It may help to first draw
right triangles in the appropriate quadrants and label
the side lengths.)
4
12
63. Suppose that sin (α ) = and sin ( β ) = ,
5
13
where 0 < α <
π
2
< β < π . Find:
(a) sin (α − β )
Find the exact value of each of the following.
(c) tan (α − β )
( )
47. cos 15
( )
64. Suppose that cos (α ) =
48. sin 75
(
49. sin 195
(b) cos (α + β )
)
(
50. cos −255
(
51. tan 105
where 0 < β < α <
)
)
π
2
8
2
and cos ( β ) = ,
17
3
. Find:
(a) sin (α + β )
(b) cos (α − β )
(c) tan (α + β )
(
52. tan 165
)
7π
53. sin
12
5π
54. cos
12
17π
55. cos −
12
11π
56. sin
12
31π
57. cos
12
65. Suppose that tan (α ) = −
where
5
1
and cos ( β ) = ,
2
2
3π
< α < β < 2π . Find:
2
(a) sin (α + β )
(b) cos (α + β )
(c) tan (α − β )
66. Suppose that tan (α ) = 3 and sin ( β ) =
where π < α <
1
,
4
3π
π
and < β < π . Find:
2
2
43π
58. sin
12
(a) sin (α − β )
13π
59. tan −
12
(c) tan (α + β )
(b) cos (α − β )
7π
60. tan
12
MATH 1330 Precalculus
91
Exercise Set 6.1: Sum and Difference Formulas
Evaluate the following.
1
1
67. cos tan −1 + tan −1
3
2
1
68. sin tan −1 ( 4 ) + tan −1
4
3
5
69. tan cos −1 − sin −1
5
13
7
4
70. tan tan −1 + cos −1
24
5
Simplify the following.
71. cos ( A ) sin ( B ) cot ( B ) + tan ( A )
72. tan ( A ) cos ( B ) cos ( A ) − cos ( A ) cot ( A ) tan ( B )
73. sin ( A − B ) sin ( A + B )
74. cos ( A + B ) cos ( A − B )
Use right triangle ABC below to answer the following
questions regarding cofunctions, in terms of side
lengths a, b, and c.
A
c
b
C
a
B
81. (a) Find sin ( A ) .
(b) Find cos ( B ) .
(c) Analyze the answers for (a) and (b). What
do you notice?
(d) What is the relationship between angles A
and B? (i.e. If you knew the measure of one
angle, how would you find the other?) Write
answers in terms of degrees.
(e) Complete the following cofunction
relationships by filling in blank. Write your
answers in terms of A.
sin ( A ) = cos ( ________ )
cos ( A ) = sin ( ________ )
Prove the following.
75. sin ( x − y ) + sin ( x + y ) = 2 sin ( x ) cos ( y )
76. cos ( x − y ) + cos ( x + y ) = 2 cos ( x ) cos ( y )
77.
cos ( x − y ) + cos ( x + y )
78.
sin ( x − y ) + sin ( x + y )
79.
sin ( x − y )
80.
sin ( x ) sin ( y )
cos ( x ) cos ( y )
sin ( x + y )
cos ( x − y )
cos ( x + y )
=
=
= 2 cot ( x ) cot ( y )
= 2 tan ( x )
tan ( x ) − tan ( y )
tan ( x ) + tan ( y )
1 + tan ( x ) tan ( y )
1 − tan ( x ) tan ( y )
82. (a) Find tan ( A ) .
(b) Find cot ( B ) .
(c) Analyze the answers for (a) and (b). What
do you notice?
(d) What is the relationship between angles A
and B? (i.e. If you knew the measure of one
angle, how would you find the other?) Write
answers in terms of degrees.
(e) Complete the following cofunction
relationships by filling in blank. Write your
answers in terms of A.
tan ( A ) = cot ( ________ )
cot ( A ) = tan ( ________ )
83. (a) Find sec ( A ) .
(b) Find csc ( B ) .
(c) Analyze the answers for (a) and (b). What
do you notice?
Continued on the next page…
92
University of Houston Department of Mathematics
Exercise Set 6.1: Sum and Difference Formulas
(d) What is the relationship between angles A
and B? (i.e. If you knew the measure of one
angle, how would you find the other?) Write
answers in terms of degrees.
97. − sec 20 sin 70
(e) Complete the following cofunction
relationships by filling in the blank. Write
your answers in terms of A.
π
5π
99. cot 2 − sec 2
12
12
sec ( A ) = csc ( _______ )
( ) ( )
π π
98. cos csc
6 3
( )
( )
100. cos 2 72 + cos 2 18
csc ( A ) = sec ( _______ )
84. Use a sum or difference formula to prove that
(
)
sin 90 − θ = cos (θ ) .
Use cofunction relationships to solve the following for
acute angle x.
( )
85. sin 75 = cos ( x )
3π
86. cos = sin ( x )
10
2π
87. sec
= csc ( x )
5
( )
88. csc 41 = sec ( x )
( )
89. tan 72 = cot ( x )
π
90. cot = tan ( x )
3
Simplify the following.
(
)
91. cos 90 − x ⋅ csc ( x )
π
92. sin − x ⋅ sec ( x )
2
π
93. sin − x ⋅ tan ( x )
2
π
94. csc − x ⋅ sin ( x )
2
(
)
(
)
95. sec 90 − θ ⋅ cos (θ )
(
96. tan 90 − x ⋅ csc 90 − x
MATH 1330 Precalculus
)
93
Exercise Set 6.2: Double-Angle and Half-Angle Formulas
Answer the following.
1.
6.
formula on tan (θ + θ ) .
π
(a) Evaluate sin 2 .
6
π
(b) Evaluate 2 sin .
6
7.
The sum formula for cosine yields the equation
cos ( 2θ ) = cos 2 (θ ) − sin 2 (θ ) . To write
cos ( 2θ ) strictly in terms of the sine function,
π
π
(c) Is sin 2 = 2sin ?
6
6
(a) Using the Pythagorean identity
cos 2 (θ ) + sin 2 (θ ) = 1 , solve for cos 2 (θ ) .
(d) Graph f ( x ) = sin ( 2 x ) and g ( x ) = 2sin ( x )
(b) Substitute the result from part (a) into the
above equation for cos ( 2θ ) .
on the same set of axes.
(e) Is sin ( 2 x ) = 2sin ( x ) ?
8.
2.
Derive the formula for tan ( 2θ ) by using a sum
π
(a) Evaluate cos 2 .
6
The sum formula for cosine yields the equation
cos ( 2θ ) = cos 2 (θ ) − sin 2 (θ ) . To write
cos ( 2θ ) strictly in terms of the cosine function,
π
(b) Evaluate 2 cos .
6
(a) Using the Pythagorean identity
cos 2 (θ ) + sin 2 (θ ) = 1 , solve for sin 2 (θ ) .
π
π
(c) Is cos 2 = 2 cos ?
6
6
(b) Substitute the result from part (a) into the
above equation for cos ( 2θ ) .
(d) Graph f ( x ) = cos ( 2 x ) and
g ( x ) = 2 cos ( x ) on the same set of axes.
(e) Is cos ( x ) = 2 cos ( x ) ?
Answer the following.
9.
Suppose that cos (α ) =
12
3π
and
< α < 2π .
2
13
Find:
3.
π
(a) Evaluate tan 2 .
6
π
(b) Evaluate 2 tan
6
π
π
(c) Is tan 2 = 2 tan ?
6
6
(d) Graph f ( x ) = tan ( 2 x ) and
g ( x ) = 2 tan ( x ) on the same set of axes.
(e) Is tan ( 2 x ) = 2 tan ( x ) ?
4.
Derive the formula for sin ( 2θ ) by using a sum
formula on sin (θ + θ ) .
(a) sin ( 2α )
(b) cos ( 2α )
(c) tan ( 2α )
10. Suppose that tan (α ) =
3
3π
and π < α <
.
4
2
Find:
(a) sin ( 2α )
(b) cos ( 2α )
(c) tan ( 2α )
11. Suppose that sin (α ) =
2
π
and < α < π .
2
5
Find:
5.
Derive the formula for cos ( 2θ ) by using a sum
(a) sin ( 2α )
formula on cos (θ + θ ) .
(b) cos ( 2α )
(c) tan ( 2α )
94
University of Houston Department of Mathematics
Exercise Set 6.2: Double-Angle and Half-Angle Formulas
12. Suppose that tan (α ) = −3 and
3π
< α < 2π .
2
(
)
(
24. sin 2 67.5 − cos 2 67.5
Find:
)
( )
1 − tan ( 41 )
−2 tan 41
(a) sin ( 2α )
25.
2
(b) cos ( 2α )
(c) tan ( 2α )
(
26. 1 − 2 cos 2 22.5
Simplify each of the following expressions as much as
possible without a calculator.
( ) ( )
13. 2 sin 15 cos 15
)
β
27. 2 cos 2 − 1
2
28.
2 tan ( 3α )
tan 2 ( 3α ) − 1
β
β
14. cos 2 − sin 2
2
2
2
( )
15. 2 cos 34 − 1
5π
16. 1 − 2 sin 2
12
(
2 tan 105
17.
2
(
)
1 − tan 105
)
7π
cos
8
( )
sin ( 67.5 )
29. (a) cos 105
(b)
18. 1 − sin 2 ( x )
7π
19. −2sin
8
x
x
The formulas for sin and cos both contain a
2
2
± sign, meaning that a choice must be made as to
whether or not the sign is positive or negative. For
each of the following examples, first state the quadrant
in which the angle lies. Then state whether the given
expression is positive or negative. (Do not evaluate the
expression.)
( )
cos (112.5 )
30. (a) sin 15
(b)
( ) ( )
20. 2 sin 23 cos 23
3π
21. cos
8
2
2 3π
− sin
8
11π
2 tan
12
22.
11π
1 − tan 2
12
7π
23. 2 sin 2
12
−1
MATH 1330 Precalculus
15π
31. (a) cos
8
13π
(b) sin
12
7π
32. (a) sin
8
19π
(b) cos
12
95
Exercise Set 6.2: Double-Angle and Half-Angle Formulas
s
In the text, tan is defined as:
2
sin ( s )
s
tan =
.
2 1 + cos ( s )
The following exercises can be used to derive this
formula along with two additional formulas for
s
tan .
2
s
33. (a) Write the formula for sin .
2
s
(b) Write the formula for cos .
2
s
(c) Derive a new formula for tan ` using the
2
sin (θ )
s
identity tan (θ ) =
, where θ = .
2
cos (θ )
Leave both the numerator and denominator
in radical form. Show all work.
34. (a) This exercise will outline the derivation for:
sin ( s )
s
. In exercise 33, it was
tan =
2 1 + cos ( s )
35. (a) In exercise 33, it was discovered that
1 − cos ( s )
s
.
tan = ±
1 + cos ( s )
2
Rationalize the numerator by multiplying
both the numerator and denominator by
1 − cos ( s ) . Simplify the expression and
s
write the new result for tan .
2
(b) A detailed analysis of the signs of the
trigonometric functions of s and s2 in
various quadrants reveals that the ± symbol
in part (a) is unnecessary. (This analysis is
lengthy and will not be shown here.) Given
this fact, rewrite the formula from part (a)
without the ± symbol. This gives yet
another formula which can be used for
s
tan .
2
(c) Use the results from Exercises 33-35 to
s
write the three formulas for tan . Which
2
formula seems easiest to use and why?
Which formula seems hardest to use and
why?
discovered that
1 − cos ( s )
s
tan = ±
.
1 + cos ( s )
2
Rationalize the denominator by multiplying
both the numerator and denominator by
1 + cos ( s ) . Simplify the expression and
s
write the result for tan .
2
(b) A detailed analysis of the signs of the
trigonometric functions of s and s2 in
various quadrants reveals that the ± symbol
in part (a) is unnecessary. (This analysis is
lengthy and will not be shown here.) Given
this fact, rewrite the formula from part (a)
without the ± symbol.
s
36. (a) In the text, tan is defined as:
2
sin ( s )
s
.
tan =
2 1 + cos ( s )
Multiply both the numerator and
denominator of the right-hand side of the
equation by 1 − cos ( s ) . Then simplify to
s
obtain a formula for tan . Show all
2
work.
(b) How does the result from part (a) compare
to the identity obtained in part (b) of
Exercise 39?
(c) How does this result from part (b) compare
with the formula given in the text
s
for tan ?
2
96
University of Houston Department of Mathematics
Exercise Set 6.2: Double-Angle and Half-Angle Formulas
Answer the following. SHOW ALL WORK. Do not
leave any radicals in the denominator, i.e. rationalize
the denominator whenever appropriate.
( ) by using a sum or difference
37. (a) Find cos 75
formula.
( ) by using a half-angle
(b) Find cos 75
formula.
(c) Enter the results from parts (a) and (b) into a
calculator and round each one to the nearest
hundredth. Are they the same?
(
)
(
)
38. (a) Find sin 165 by using a sum or difference
formula.
(b) Find sin 165 by using a half-angle
formula.
(c) Enter the results from parts (a) and (b) into a
calculator and round each one to the nearest
hundredth. Are they the same?
(
)
(
)
(
)
Find the exact value of each of the following by using a
half-angle formula. Do not leave any radicals in the
denominator, i.e. rationalize the denominator
whenever appropriate.
13π
41. (a) sin
12
13π
(b) cos
12
13π
(c) tan
12
3π
42. (a) sin
8
3π
(b) cos
8
3π
(c) tan
8
11π
43. (a) sin
8
11π
(b) cos
8
11π
(c) tan
8
39. (a) Find sin 112.5 by using a half-angle
formula.
(b) Find cos 112.5 by using a half-angle
formula.
(c) Find tan 112.5 by computing
(
).
cos (112.5 )
Find tan (112.5 ) by using a half-angle
sin 112.5
(d)
(b)
(c)
formula.
(
)
40. (a) Find sin 105 by using a half-angle
(b)
(
(b) Find cos 105
) by using a half-angle
(c)
formula.
(
)
(
)
(c) Find tan 105 by computing
MATH 1330 Precalculus
( )
cos ( 285 )
tan ( 285 )
( ).
cos (105 )
sin 105
(d) Find tan 105 by using a half-angle
formula.
45. (a) sin 285
formula.
(
)
cos (157.5 )
tan (157.5 )
44. (a) sin 157.5
17π
46. (a) sin
12
17π
(b) cos
12
17π
(c) tan
12
97
Exercise Set 6.2: Double-Angle and Half-Angle Formulas
θ
(g) Find the exact value of cos .
2
Answer the following.
47. If cos (θ ) =
5
3π
and
< θ < 2π ,
2
9
θ
(h) Find the exact value of tan .
2
(a) Determine the quadrant of the terminal side
of θ .
(b) Complete the following: ___ <
θ
2
< ___
49. If tan (θ ) = −
(c) Determine the quadrant of the terminal side
of
θ
2
θ
(a) Find the exact value of sin .
2
.
θ
(b) Find the exact value of cos .
2
(d) Based on the answer to part (c), determine
θ
the sign of sin .
2
(e) Based on the answer to part (c), determine
θ
the sign of cos .
2
θ
(c) Find the exact value of tan .
2
θ
(a) Find the exact value of sin .
2
θ
(g) Find the exact value of cos .
2
θ
(b) Find the exact value of cos .
2
θ
(h) Find the exact value of tan .
2
21
3π
and π < θ <
,
2
5
(a) Determine the quadrant of the terminal side
of θ .
θ
(c) Find the exact value of tan .
2
Prove the following.
51.
1 − cos ( 2 x )
52.
cos ( 3 x )
(b) Complete the following:
_____ <
θ
< _____
2
(c) Determine the quadrant of the terminal side
of
θ
2
.
sin ( 2 x )
cos ( x )
−
= tan ( x )
sin ( 3 x )
sin ( x )
= −2
53. 1 − 2 sin ( x ) 1 + 2 sin ( x ) = cos ( 2 x )
54. cos 4 ( x ) − sin 4 ( x ) = cos ( 2 x )
(d) Based on the answer to part (c), determine
θ
the sign of sin .
2
x
55. csc ( x ) − cot ( x ) = tan
2
(e) Based on the answer to part (c), determine
θ
the sign of cos .
2
x
1 + tan 2
2 = sec θ
56.
( )
x
1 − tan 2
2
θ
(f) Find the exact value of sin .
2
98
5
3π
and
< θ < 2π ,
2
6
50. If cos (θ ) =
θ
(f) Find the exact value of sin .
2
48. If sin (θ ) = −
7
π
and < θ < π ,
2
3
University of Houston Department of Mathematics
Exercise Set 6.3: Solving Trigonometric Equations
Does the given x-value represent a solution to the
given trigonometric equation? Answer yes or no.
1.
x=
π
6
; 2 cos 2 ( x ) − 7 cos ( x ) = 4
2.
5π
x=
; 3 tan 2 ( x ) = 8 tan ( x ) − 5
4
3.
x=
4.
3π
; sin 3 ( x ) − 4sin 2 ( x ) − 2sin ( x ) = −3
2
x=−
π
3
; 6 cos 2 ( x ) − 7 cos ( x ) − 2 = 0
For each of the following equations,
(a) Solve the equation for 0 ≤ x < 360
(b) Solve the equation for 0 ≤ x < 2π .
(c) Find all solutions to the equation, in radians.
5.
cos ( x ) = −
6.
sin ( x ) =
1
2
3
2
7.
tan ( x ) = −1
8.
cos ( x ) = 0
9.
sin ( x ) = 1
10.
When solving an equation such as sin ( x ) = − 12
for 0 ≤ x < 360 , a typical thought process is to first
compute sin −1 ( 12 ) to find the reference angle (in this
case, 30 ) and then use that reference angle to find
solutions in quadrants where the sine value is negative
(in this case, 210 and 330 . In each of the the
following examples of the form sin ( x ) = C ,
(a) Use a calculator to find sin −1 C to the nearest
tenth of a degree. This represents the
reference angle.
(b) Use the reference angle from part (a) to find
the solutions to the equation for
0 ≤ x < 360 .
19. sin ( x ) = −
4
5
20. sin ( x ) = −0.3
21. sin ( x ) = −0.46
22. sin ( x ) = −
2
5
The method in Exercises 19-22 can be used for other
trigonometric functions as well. Use a calculator to
solve each of the following equations for 0 ≤ x < 360 .
Round answers to the nearest tenth of a degree. If no
solution exists, state “No solution.”
3 cot ( x ) = −1
2 3
11. sec ( x ) = −
3
12. csc ( x ) = −2
Note: Since calculators do not contain inverse keys for
cosecant, secant, and cotangent, use reciprocal
relationships to rewrite equations in terms of sine, cosine
and tangent. For example, to solve the equation
sec ( x ) = −4 , first rewrite the equation as cos ( x ) = − 14 and
then proceed to solve the equation.
13. 2 cos ( x ) = 2
23. (a) cos ( x ) = −
14. sin ( x ) = 2
15. 2sin 2 ( x ) − 5sin ( x ) − 3 = 0
16. 2 cos 2 ( x ) + cos ( x ) = 1
17. cos 2 ( x ) = 2 cos ( x ) − 1
3
7
(b) csc ( x ) =
7
5
24. (a) tan ( x ) = −6
(b) sec ( x ) = −7
25. (a) cot ( x ) = −2.9
(b) sin ( x ) = 5.6
26. (a) csc ( x ) =
1
4
(b) tan ( x ) = 2.5
18. 2sin 2 ( x ) = −7 sin ( x ) + 4
MATH 1330 Precalculus
99
Exercise Set 6.3: Solving Trigonometric Equations
Answer the following.
27. The following example is designed to
demonstrate a common error in solving
trigonometric equations. Consider the equation
2 cos 2 ( x ) = 3 cos ( x ) for 0 ≤ x < 2π .
(a) Divide both sides of the equation by cos ( x )
and then solve for x.
(b) Move all terms to the left side of the
equation, and then solve for x.
(c) Are the answers in parts (a) and (b) the
same?
(d) Which method is correct, part (a) or (b)?
Why is the other method incorrect?
28. The following example is designed to
demonstrate a common error in solving
trigonometric equations. Consider the equation
tan 2 ( x ) = tan ( x ) for 0 ≤ x < 360 .
The following exercises show a method of solving an
equation of the form:
sin ( Ax + B ) = C , for 0 ≤ x < 2π .
(The same method can be used for the other five
trigonometric functions as well, and can similarly be
applied to intervals other than 0 ≤ x < 2π .) Answer
the following, using the method described below.
(a) Write the new interval obtained by
multiplying each term in the solution interval
(in this case, 0 ≤ x < 2π ) by A and then
adding B.
(b) Let u = Ax + B . Find all solutions to
sin ( u ) = C within the interval obtained in
part (a).
(c) For each solution u from part (b), set up and
solve u = Ax + B for x. These x-values
represent all solutions to the initial equation.
37. sin ( 2 x ) =
(a) Divide both sides of the equation by tan ( x )
and then solve for x.
38. sin ( 3x ) = −
(b) Move all terms to the left side of the
equation, and then solve for x.
(c) Are the answers in parts (a) and (b) the
same?
(d) Which method is correct, part (a) or (b)?
Why is the other method incorrect?
Solve the following equations for 0 ≤ x < 360 . If no
solution exists, state “No solution.”
29.
2 sin 2 ( x ) = sin ( x )
2
2
1
2
π
3
39. sin x − = −
2
2
π
2
40. sin x + = −
4
2
41. sin ( 3x + π ) = 0
x π 1
42. sin − =
2 3 2
30. cos 2 ( x ) = − cos ( x )
31.
3 tan 2 ( x ) = − tan ( x )
32. 2sin 2 ( x ) = sin ( x )
33. 4sin 2 ( x ) cos ( x ) − cos ( x ) = 0
34. sin ( x ) tan 2 ( x ) = sin ( x )
100
Solve the following, using either the method above or
the method described in the text. If no solution exists,
state “No solution.”
43. 2 cos ( 2 x ) = 2 , for 0 ≤ x < 2π
44.
3 tan ( 2 x ) + 1 = 0 , for 0 ≤ x < 360
35. 2 cos3 ( x ) = −3cos 2 ( x ) − cos ( x )
45. csc ( 2 x ) = − 2 , for 0 ≤ x < 360
36. 2sin 3 ( x ) + 9sin 2 ( x ) = 5sin ( x )
46. sec ( 2 x ) = 2 , for 0 ≤ x < 2π
University of Houston Department of Mathematics
Exercise Set 6.3: Solving Trigonometric Equations
x
47. 2sin = −1 , for 0 ≤ x < 2π
2
Use a calculator to solve each of the following
equations for 0 ≤ x < 360 . Round answers to the
nearest tenth of a degree. If no solution exists, state
“No solution.”
48. tan ( 3x ) = 3 , for 0 ≤ x < 360
63. 15sin 2 ( x ) = 6sin ( x )
49. 2sin ( 3x ) + 3 = 0 , for 0 ≤ x < 360
64. 6 tan 2 ( x ) − tan ( x ) = 1
x
50. 2 cos = 3 , for 0 ≤ x < 360
2
65. 3cot 2 ( x ) = 5cot ( x ) + 2
51. 2 cos ( 4 x ) = − 3 , for 0 ≤ x < 2π
66. 4 csc 2 ( x ) − 8csc ( x ) = −3
52. csc ( 5 x ) = −1 , for 0 ≤ x < 2π
67. sec2 ( x ) = 16
(
)
53. tan x + 90 = 1 , for 0 ≤ x < 360
(
)
68. 3cos 2 ( x ) = −2 cos ( x )
54. 2 cos x − 45 = −1 , for 0 ≤ x < 360
55.
5π
3 sec x −
6
+ 2 = 0 , for 0 ≤ x < 2π
56. cot ( x + π ) = 0 , for 0 ≤ x < 2π
(
Solve the following for 0 ≤ x < 2π . If no solution
exists, state “No solution.”
69. 2sin 2 ( x ) − cos ( x ) = 1
70. 6 cos 2 x − 13sin ( x ) − 11 = 0
57. sin 2 x − 270 = 1 , for 0 ≤ x < 360
)
71. cot 2 ( x ) + csc ( x ) − 1 = 0
58. tan ( 3x + 2π ) = 0 , for 0 ≤ x < 2π .
72. sec2 ( x ) + 2 tan ( x ) = 0
x
59. tan + 60 = 1 , for 0 ≤ x < 360
2
73. tan ( x ) = 3
x π
60. cos − + 1 = 0 , for 0 ≤ x < 2π .
3 3
(
)
61. 2 cos 45 − 2 x = 3 , for 0 ≤ x < 360
π
62. 2sin −3x − = 2 , for 0 ≤ x < 2π .
4
74. sec ( x ) = 1
75. log 2 cos ( x ) = −1
1
76. log3 cot ( x ) =
2
77. log3 tan ( x ) = 0
78. log
MATH 1330 Precalculus
2
csc ( x ) = 1
101
Exercise Set 7.1: Solving Right Triangles
For each of the following examples,
(a) Write a trigonometric equation that can be
used to find x, and write a trigonometric
equation that can be used to find y. Each
equation should involve the given acute angle
and the given side length from the diagram.
(Do not solve the equations in this step.)
6.
y
x
30o
15
7.
(b) Solve the equations from part (a) to find exact
values for x and y.
x
45
7 2
(c) As an alternative solution to the method
above, use properties of special right triangles
(from Section 4.1) to find exact values of x and
y.
1.
y
8.
10
45o
y
11
x
30o
x
y
2.
45o
y
x
Use a calculator to evaluate the following. Round
answers to the nearest ten-thousandth.
( )
(b) cos 73
10. (a) cos 16
( )
(b) tan 67
π
11. (a) tan
9
2π
(b) sin
7
π
12. (a) sin
10
3π
(b) cos
11
9.
7
2
3.
y
7 3
60o
x
(a) sin 48
( )
( )
4.
In the equations below, x represents an acute angle of
a triangle. Use a calculator to solve for x. Round
answers to the nearest hundredth of a degree.
x
y
13. sin ( x ) = 0.7481
60o
9
14. cos ( x ) = 0.0256
5.
15. tan ( x ) = 4
y
5
45o
16. tan ( x ) =
8
15
x
102
University of Houston Department of Mathematics
Exercise Set 7.1: Solving Right Triangles
17. cos ( x ) =
16
17
25. In ∆GEO with right angle E, GE = 9 cm and
OE = 7 cm . Find ∠G .
18. sin ( x ) =
4
7
26. In ∆CAR with right angle R, CA = 11 in and
AR = 8 in . Find ∠A .
Find the indicated side or angle measures (where
angles are measured in degrees). Round answers to the
nearest hundredth.
Solve each of the following triangles, i.e. find all
missing side and angle measures (where angles are
measured in degrees). Round all answers to the
nearest hundredth.
19. Find x.
J
27.
23
38o
3
x
61
M
20. Find x.
D
54o
x
28.
C
14
7
21. In ∆ABC with right angle C, ∠A = 74 and
BC = 32 in . Find AC .
U
B
12
22. In ∆QRS with right angle R, QR = 7 in and
∠Q = 12 . Find QS .
29.
P
8
3
23. Find ∠H .
H
H
L
17
I
13
J
T
30.
8
24. Find θ .
47
4
I
θ
R
3
MATH 1330 Precalculus
103
Exercise Set 7.1: Solving Right Triangles
For each of the following examples,
(a) Draw a diagram to represent the given
situation.
(b) Find the indicated measure, to the nearest
tenth.
31. An isosceles triangle has sides measuring 9
inches, 14 inches, and 14 inches. What are the
measures of its angles?
32. An isosceles triangle has sides measuring 10
inches, 7 inches, and 7 inches. What are the
measures of its angles?
33. An 8-foot ladder is leaned against the side of the
house. If the ladder makes a 72 angle with the
ground, how high up does the ladder reach?
34. A 14-foot ladder is leaned against the side of the
house. If the ladder makes a 63 angle with the
ground, how high up does the ladder reach?
35. The angle of elevation to the top of a flagpole
from a point on the ground 40 ft from the base of
the flagpole is 51 . Find the height of the
flagpole.
36. The angle of elevation to the top of a flagpole
from a point on the ground 18 meters from the
base of the flagpole is 43 . Find the height of the
flagpole.
37. A loading dock is 8 feet high. If a ramp makes an
angle of 34 with the ground and is attached to
the loading dock, how long is the ramp?
38. A loading dock is 5 feet high. If a ramp makes an
angle of 25 with the ground and is attached to
the loading dock, how long is the ramp?
41. Jeremy is flying his kite at Lake Michigan and
lets out 200 feet of string. If his eyes are 6 feet
from the ground as he sights the kite at an 55
angle of elevation, how high is the kite?
(Assume that he is holding the kite near his
eyes.)
42. Pam is flying a kite at the beach and lets out 120
feet of string. If her eyes are 5 feet from the
ground as she sights the kite at a 64 angle of
elevation, how high is the kite? (Assume that she
is also holding the kite near her eyes.)
43. Doug is out hiking and walks to the edge of a
cliff to enjoy the view. He looks down and sights
a cougar at a 41 angle of depression. If the cliff
is 60 meters high, approximately how far is the
cougar from the base of the cliff? (Disregard
Doug’s height in your calculations.)
44. Juliette wakes up in the morning and looks out
her window into the back yard. She sights a
squirrel at a 54 angle of depression. If Juliette
is looking out the window at a point which is 28
feet from the ground, how far is the squirrel from
the base of the house?
45. From a point on the ground, the angle of
elevation to the top of a mountain is 42 .
Moving out a distance of 150 m (on a level
plane) to another point on the ground, the angle
of elevation is 33 . Find the height of the
mountain.
46. From a point on the ground, the angle of
elevation to the top of a mountain is 31 .
Moving out a distance of 200 m (on a level
plane) to another point on the ground, the angle
of elevation is 18 . Find the height of the
mountain.
39. Silas stands 600 feet from the base of the Sears
Tower and sights the top of the tower. If the
Sears Tower is 1,450 feet tall, approximate the
angle of elevation from Silas’ perspective as he
sights the top of the tower. (Disregard Silas’
height in your calculations.)
40. Mimi is sitting in her boat and sees a lighthouse
which is 20 meters tall. If she sights the top of
the lighthouse at a 32 angle of elevation,
approximately how far is Mimi’s boat from the
base of the lighthouse? (Disregard Mimi’s
distance from the water in your calculations.)
104
University of Houston Department of Mathematics
Exercise Set 7.2: Area of a Triangle
Use the formula A = 12 bh to find the area of each of
7.
the following triangles. (You may need to find the base
and/or the height first, using trigonometric ratios or
the Pythagorean Theorem.) Give exact values
whenever possible. Otherwise, round answers to the
nearest hundredth.
Note: Figures may not be drawn to scale.
57o
9 in
1.
8.
38o
8 cm
10 cm
7m
2.
7 in
In ∆ABC below, AD ⊥ BC . Use the diagram below
to answer the following questions.
A
3 in
b
c
h
8 ft
3.
C
3 ft
D
B
a
9.
4.
(a) Use ∆ACD to write a trigonometric ratio
that involves ∠C , b, and h.
(b) Using the equation from part (a), solve for h.
61 cm
6 cm
(c) If a represents the base of ∆ABC , and h
represents the height, then the area K of
∆ABC is K = 12 ah . Substitute the equation
from part (b) into this equation to obtain a
formula for the area of ∆ABC that no
longer contains h.
5.
10 m
10. (a) Use ∆ABD to write a trigonometric ratio
that involves ∠B , c, and h.
30o
(b) Using the equation from part (a), solve for h.
6.
8m
45o
MATH 1330 Precalculus
(c) If a represents the base of ∆ABC , and h
represents the height, then the area K of
∆ABC is K = 12 ah . Substitute the equation
from part (b) into this equation to obtain a
formula for the area of ∆ABC that no
longer contains h.
105
Exercise Set 7.2: Area of a Triangle
Find the area of each of the following triangles. Give
exact values whenever possible. Otherwise, round
answers to the nearest hundredth.
Note: Figures may not be drawn to scale.
C
16.
50o
8 cm
4 cm
B
11.
120o
6 in
A
10 in
A
C
12.
T
17.
J
53o
B
4.1 m
11 cm
36o
30o
A
13.
C
10 cm
5.6 m
M
A
S
18.
L
7.9 m
21o
9m
125o
Q
45o
J
3.5 m
R
K
8m
14.
F
5 ft
Answer the following. Give exact values whenever
possible. Otherwise, round answers to the nearest
hundredth.
135o
D
19. Find the area of an isosceles triangle with legs
measuring 7 inches and base angles measuring
22.5 .
E
7 ft
15.
G
12 m
28o
D
8m
O
20. Find the area of an isosceles triangle with legs
measuring 10 cm and base angles measuring
75 .
21. Find the area of ∆GHJ , where ∠G = 120 ,
h = 8 cm , and j = 15 cm .
22. Find the area of ∆PQR , where ∠R = 47 ,
p = 7 in , and q = 10 in .
106
University of Houston Department of Mathematics
Exercise Set 7.2: Area of a Triangle
23. Find the area of ∆TUV , where ∠T = 28 ,
∠U = 81 , t = 12 m , and u = 6.5 m .
24. Find the area of ∆FUN , where ∠F = 92 ,
∠N = 28 , f = 9 cm , and n = 8 cm .
In the following questions, the radius of circle O is
given, as well as the measure of central angle AOB.
Find the area of the segment of circle O bounded by
AB and AB . Give exact values whenever possible.
Otherwise, round answers to the nearest hundredth.
25. (a) Find all solutions to sin ( x ) = 0.5 for
O
0 < x < 180 .
(b) If the area of ∆RST is 20 cm2, r = 16 cm
and t = 5 cm , find all possible measures for
∠S .
26. (a) Find all solutions to sin ( x ) = 0.2 for
0 < x < 180 .
B
A
37. Radius: 8 cm
Central Angle:
3π
7
(b) If the area of ∆PHL is 12 m2, p = 10 m and
h = 6 m , find all possible measures for ∠L .
2
38. Radius: 7 in
Central Angle:
27. If the area of ∆TRG is 37 in , t = 8 in and
g = 12.5 in , find all possible measures for ∠R .
28. If the area of ∆PSU is 20 2 cm2, p = 4 cm
and s = 5 cm , find all possible measures for
∠U .
29. A regular octagon is inscribed in a circle of
radius 12 in. Find the area of the octagon.
4π
5
39. Radius: 4 in
Central Angle:
π
3
40. Radius: 6 cm
Central Angle:
3π
4
30. A regular hexagon is inscribed in a circle of
radius 11 cm. Find the area of the hexagon.
31. A regular hexagon is circumscribed about a
circle of radius 6 in. Find the area of the
hexagon.
32. A regular hexagon is circumscribed about a
circle of radius 10 cm. Find the area of the
hexagon.
33. Find the area of an equilateral triangle with side
length 4 ft.
34. Find the area of a regular hexagon with side
length 10 cm.
35. Find the area of a regular decagon with side
length 8 in.
36. Find the area of a regular octagon with side
length 12 m.
MATH 1330 Precalculus
107
Exercise Set 7.3: The Law of Sines and the Law of Cosines
Find the indicated part of each triangle. Give exact
values whenever possible. Otherwise, round answers
to the nearest hundredth. If no such triangle exists,
state “No such triangle exists.”
1.
∆ABC
a=9m
c=5m
7.
∠N = 63
∠S = 48
s=?
8.
∠B = 71
b=?
2.
3.
∆ABC
a = 9 in
∠A = 60
∠C = 45
c=?
4.
6.
108
9.
∆CAN
c = 11 cm
a = 9 cm
n = 14 cm
∠C = ?
10. ∆HAT
a = 14 mm
∠A = 45
∠T = 15
h=?
∆RST
s = 8 mm
∠S = 135
∠T = 30
t =?
5.
∆PSU
s = 2.5 in
∠P = 112
∠U = 37
p=?
∆ABC
a = 16 cm
∠A = 35
∠B = 65
b=?
∆SUN
n = 10 cm
11. ∆RST
r = 9 mm
s = 15 in
t = 4 in
∠R = ?
∆NAP
n = 5 in
a = 10 in
p = 7 in
∠N = ?
12. ∆BOX
o = 10 cm
x = 7 cm
∆KLM
k = 3 in
l = 4 in
m = 8 in
∠M = ?
13. ∆TRY
y = 12 cm
∠B = 43
b=?
∠T = 30
∠R = 105
t =?
University of Houston Department of Mathematics
Exercise Set 7.3: The Law of Sines and the Law of Cosines
14. ∆WIN
w = 4.9 ft
20. ∆ABC
b = 10 ft
∠I = 42
∠N = 100
n=?
Find all possible measures for the indicated angle of
the triangle. (There may be 0, 1, or 2 triangles with the
given measures.) Round answers to the nearest
hundredth. If no such triangle exists, state “No such
triangle exists.”
15. ∆DEF
d = 25 mm
e = 13 mm
c = 7 ft
∠B = 70
∠A = ?
21. ∆ANT
n = 12 cm
t = 15 cm
∠T = 129
∠A = ?
22. ∆JAY
j = 15 in
y = 9 in
∠E = 21
∠D = ?
16. ∆CAR
c = 3 cm
r = 7 cm
∠C = 15
∠R = ?
∠Y = 160
∠J = ?
Solve each of the following triangles, i.e. find all
missing side and angle measures. There may be two
groups of answers for some exercises. Round answers
to the nearest hundredth. If no such triangle exists,
state “No such triangle exists.”
Note: Figures may not be drawn to scale.
17. ∆ABC
b = 9 cm
c = 6 cm
F
23.
35o
∠B = 95
∠C = ?
125o
D
18. ∆BUS
b = 18 mm
u = 16 mm
∠B = 120
∠U = ?
24.
9.42 ft
C
50o
8 cm
4 cm
19. ∆GEO
g = 11 cm
o = 4 cm
∠O = 33
∠G = ?
E
A
T
25. ∆WHY
h = 8 in
y = 4 in
∠Y = 28
MATH 1330 Precalculus
109
Exercise Set 7.3: The Law of Sines and the Law of Cosines
26. ∆ABC
a = 12 in
34. ∆SEA
s = 8 cm
e = 9 cm
a = 21 cm
b = 17 in
c = 6 in
27. ∆RUN
r = 7 cm
u = 12 cm
n = 4 cm
Find the area of each of the following quadrilaterals.
Round all intermediate computations to the nearest
hundredth, and then round the area of the
quadrilateral to the nearest tenth.
Note: Figures may not be drawn to scale.
28. ∆JKL
j=6m
k =5m
U
35.
25 cm
∠J = 74
128o
29. ∆PEZ
p = 12 mm
z = 10 mm
17 cm
∠E = 130
30. ∆BAD
b = 11 ft
T
30 cm
S
O
36.
a = 9 ft
d = 5 ft
C
18 cm
8 cm
31. ∆PIG
p = 6 cm
i = 13 cm
g = 11 cm
R
49o
G
23 cm
U
U
37.
Q
32. ∆QIZ
i = 12.4 mm
z = 11.5 mm
∠Z = 66
15 in
12 in
122o
D
115o
17 in
33. ∆DEF
d = 10 mm
f = 3 mm
C
38.
20 in
A
123o
19 in
14 in
∠F = 21
E
110
A
33 in
K
University of Houston Department of Mathematics
Exercise Set 7.3: The Law of Sines and the Law of Cosines
Answer the following. Round answers to the nearest
hundredth.
Note: Figures may not be drawn to scale.
39. (a) Use the Law of Sines or the Law of Cosines
to find the measures of the acute angles of a
3-4-5 right triangle.
(b) Use right triangle trigonometry to find the
measures of the acute angles of a 3-4-5 right
triangle.
40. (a) Use the Law of Sines or the Law of Cosines
to find the measures of the acute angles of a
7-24-25 right triangle.
(b) Use right triangle trigonometry to find the
measures of the acute angles of a 7-24-25
right triangle.
5
y . Find the
4
41. In ∆HEY , ∠H = 125 and h =
measures of ∠E and ∠Y .
42. In ∆FUN , ∠F = 100 and u =
2
f . Find the
3
measures of ∠U and ∠N .
43. Find the length of AD .
A
11 mm
7 mm
B
6 mm
D
C
8 mm
44. Find the length of AD .
A
14 mm
6 mm
B
8 mm
MATH 1330 Precalculus
D
5 mm
C
111
Exercise Set 8.1: Parabolas
Write each of the following equations in the standard
form for the equation of a parabola, where the
standard form is represented by one of the following
equations:
13.
( x − 2 )2 = 8 ( y + 5 )
14.
( y − 4 )2 = 16 x
( x − h)2 = 4 p ( y − k )
2
( y − k ) = 4 p ( x − h)
15. y 2 = 4 ( x − 3)
( x − h ) 2 = −4 p ( y − k )
2
( y − k ) = −4 p ( x − h )
16.
( x + 3)2 = 4 ( y − 1)
17. ( x + 5)2 = −2 ( y − 4 )
1.
y 2 − 14 y − 2 x + 43 = 0
2.
x 2 + 10 x − 12 y = −61
18.
( y + 1)2 = −10 ( x + 3)
3.
−9 y = x 2 − 8 x − 10
19.
( y − 6 ) 2 = −1 ( x − 2 )
4.
−7 x = y 2 − 10 y + 24
20.
( x − 1)2 = −8 ( y − 6 )
5.
x = 3 y 2 − 24 y + 50
21. x 2 + 12 x − 6 y + 24 = 0
6.
y = 2 x 2 + 12 x + 15
22. x 2 − 2 y = 8 x − 10
7.
3x 2 − 3x − 5 − y = 0
23. y 2 − 8 x = 4 y + 36
8.
5 y2 + 5 y = x − 6
24. y 2 + 6 y − 4 x + 5 = 0
25. x 2 + 25 = −16 y − 10 x
For each of the following parabolas,
(a) Write the given equation in the standard form
for the equation of a parabola. (Some
equations may already be given in standard
form.)
It may be helpful to begin sketching the graph for
part (h) as a visual aid to answer the questions
below.
26. y 2 + 10 y + x + 28 = 0
27. y 2 − 4 y + 2 x − 4 = 0
28. 12 y + x 2 − 4 x = −16
29. 3 y 2 + 30 y − 8 x + 67 = 0
30. 5 x 2 + 30 x − 16 y = 19
(b) State the equation of the axis.
(c)
(d)
(e)
(f)
(g)
State the coordinates of the vertex.
State the equation of the directrix.
State the coordinates of the focus.
State the focal width.
State the coordinates of the endpoints of the
focal chord.
(h) Sketch a graph of the parabola which includes
the features from (c)-(e) and (g). Label the
vertex V and the focus F.
31. 2 x 2 − 8 x + 7 y = 34
32. 4 y 2 − 8 y + 9 x + 40 = 0
Use the given features of each of the following
parabolas to write an equation for the parabola in
standard form.
33. Vertex:
Focus:
9.
( −2, 5 )
( 4, 5 )
x2 − 4 y = 0
10. y 2 − 12 x = 0
34. Vertex: (1, − 3)
Focus: (1, 0 )
11. −10x = y 2
12. −6 y = x 2
112
University of Houston Department of Mathematics
Exercise Set 8.1: Parabolas
35. Vertex:
Focus:
36. Vertex:
Focus:
37. Focus:
( 2, 0 )
48. Endpoints of focal chord:
( −4, − 2 )
( −6, − 2 )
Answer the following.
49. Write an equation of the line tangent to the
parabola with equation f ( x ) = x 2 + 5 x + 4 at:
( −2, − 3)
(a) x = 3
(b) x = −2
( 4, 1)
Directrix: y = 5
39. Focus:
50. Write an equation of the line tangent to the
parabola with equation f ( x ) = −3x 2 + 6 x + 1 at:
( 4, − 1)
(a) x = 0
(b) x = −1
Directrix: x = 7 12
40. Focus:
( −3, 5)
51. Write an equation of the line tangent to the
parabola with equation f ( x ) = 2 x 2 + 5 x − 1 at:
Directrix: x = −4
41. Focus:
(a) x = −1
1
(b) x =
2
( −4, − 2 )
Opens downward
p=7
52. Write an equation of the line tangent to the
parabola with equation f ( x ) = − x 2 + 4 x − 2 at:
42. Focus: (1, 5 )
Opens to the right
p=3
43. Vertex:
(a) x = −3
3
(b) x =
2
( 5, 6 )
Opens upward
Length of focal chord: 6
53. Write an equation of the line tangent to the
parabola with equation f ( x ) = 4 x 2 − 5 x − 3 at
the point (1, − 4 ) .
1
44. Vertex: 0,
2
Opens to the left
Length of focal chord: 2
45. Vertex:
54. Write an equation of the line tangent to the
parabola with equation f ( x ) = 2 x 2 + 5 x + 4 at
the point ( −3, 7 ) .
( −3, 2 )
Horizontal axis
Passes through ( 6, 5 )
46. Vertex:
and ( 6, 3)
Opens downward
( 2, − 4 )
Directrix: y = −9
38. Focus:
( −2, 3)
55. Write an equation of the line tangent to the
parabola with equation f ( x ) = − x 2 − 6 x + 1 at
the point ( −5, 6 ) .
( 2,1)
Vertical axis
Passes through ( 8, 5)
47. Endpoints of focal chord:
56. Write an equation of the line tangent to the
parabola with equation f ( x ) = −3x 2 + 4 x + 9 at
( 0, 5)
and ( 0, − 5 )
the point ( 2, 5 ) .
Opens to the left
MATH 1330 Precalculus
113
Exercise Set 8.1: Parabolas
Give the point(s) of intersection of the parabola and
the line whose equations are given.
57. f ( x ) = x 2 − 4 x + 11
g ( x ) = 5x − 3
58. f ( x ) = x 2 − 8 x + 1
g ( x ) = −2 x − 10
59. f ( x ) = x 2 + 10 x + 10
g ( x ) = 8x + 9
60. f ( x ) = x 2 − 5 x − 10
g ( x ) = −7 x + 14
61. f ( x ) = − x 2 + 6 x − 5
g ( x ) = 6 x − 14
62. f ( x ) = − x 2 + 3x − 2
g ( x ) = −5 x + 13
63. f ( x ) = −3x 2 + 6 x + 1
g ( x ) = −3 x + 7
64. f ( x ) = −2 x 2 + 8 x − 5
g ( x) = 6x − 5
114
University of Houston Department of Mathematics
Exercise Set 8.2: Ellipses
Write each of the following equations in the standard
form for the equation of an ellipse, where the standard
form is represented by one of the following equations:
( x − h)2 ( y − k )2
+
a2
+
b2
2
=1
( x − h)2 ( y − k )2
+
b2
+
a2
=1
2
1.
25 x + 4 y − 100 = 0
2.
9 x 2 + 16 y 2 − 144 = 0
3.
9 x 2 − 36 x + 4 y 2 − 32 y + 64 = 0
4.
4 x 2 + 24 x + 16 y 2 − 32 y − 12 = 0
5.
3x 2 + 2 y 2 − 30 x − 12 y = −87
(e) State the coordinates of the foci.
(f) State the eccentricity.
(g) Sketch a graph of the ellipse which includes
the features from (b)-(e). Label the center C,
and the foci F1 and F2.
11.
x2 y2
+
=1
9 49
12.
x2 y2
+
=1
36 4
13.
( x − 2)
2
+
16
6.
x 2 + 8 y 2 + 113 = 14 x + 48 y
7.
16 x 2 − 16 x − 64 = −8 y 2 − 24 y + 42
2
x 2 ( y + 1)
+
=1
14.
9
5
15.
8.
10. The sum of the focal radii of an ellipse is always
equal to __________.
17.
It may be helpful to begin sketching the graph for
part (g) as a visual aid to answer the questions
below.
(b) State the coordinates of the center.
(c) State the coordinates of the vertices of the
major axis, and then state the length of the
major axis.
(d) State the coordinates of the vertices of the
minor axis, and then state the length of the
minor axis.
MATH 1330 Precalculus
+
( x + 5)
2
+
( x + 4)
2
+
9
18.
( x + 2)
2
36
19.
+
( y + 3)
2
=1
16
( y + 2)
2
=1
25
( y − 3)
2
=1
1
( y + 3)
2
16
( x − 2 )2 ( y + 4 )2
+
11
20.
Answer the following for each ellipse. For answers
involving radicals, give exact answers and then round
to the nearest tenth.
(a) Write the given equation in the standard form
for the equation of an ellipse. (Some equations
may already be given in standard form.)
2
16
Answer the following.
(a) What is the equation for the eccentricity, e,
of an ellipse?
(b) As e approaches 1, the ellipse appears to
become more (choose one):
elongated
circular
(c) If e = 0 , the ellipse is a __________.
( x − 2)
25
18 x 2 + 9 y 2 = 153 − 24 x + 6 y
16.
9.
y2
=1
4
( x + 3)
20
2
+
36
( y − 5)
=1
=1
2
4
=1
21. 4 x 2 + 9 y 2 − 36 = 0
22. 4 x 2 + y 2 = 1
23. 25 x 2 + 16 y 2 − 311 = 50 x − 64 y
24. 16 x 2 + 25 y 2 = 150 y + 175
25. 16 x 2 − 32 x + 4 y 2 − 40 y + 52 = 0
26. 25 x 2 + 9 y 2 − 100 x + 54 y − 44 = 0
27. 16 x 2 + 7 y 2 + 64 x − 42 y + 15 = 0
28. 4 x 2 + 3 y 2 − 16 x + 6 y − 29 = 0
115
Exercise Set 8.2: Ellipses
Use the given features of each of the the following
ellipses to write an equation for the ellipse in standard
form.
29. Center:
( 0, 0 )
a =8
b=5
Horizontal Major Axis
38. Foci:
( 0, 0 )
a=7
b=3
Vertical Major Axis
31. Center:
( −4, 7 )
a=5
b=3
Vertical Major Axis
32. Center:
( 2, − 4 )
a =5
b=2
Horizontal Major Axis
39. Foci:
( −3, − 5)
Length of major axis = 6
Length of minor axis = 4
Horizontal Major Axis
34. Center:
( 2,1)
and ( 7, 2 )
( −3, 4 )
and ( 7, 4 )
Passes through the point ( 2,1)
41. Center:
( −5, 2 )
a =8
3
e=
4
Vertical major axis
42. Center:
( −4, − 2 )
a=6
2
e=
3
Horizontal major axis
e=
( 0, 4 )
and ( 0, 8 )
1
3
44. Foci: (1, 5 ) and (1, − 3)
e=
Length of major axis = 10
Length of minor axis = 2
Vertical Major Axis
( −1, 2 )
Passes through the point ( 3, 5)
43. Foci:
33. Center:
and ( −2, 5 )
a =8
40. Foci:
30. Center:
( −2, −3)
1
2
45. Foci:
( 2, 3)
and ( 6, 3)
e = 0.4
35. Foci:
( 2, 5 )
and ( 2, − 5 )
a =9
46. Foci:
( 2,1)
and (10, 1)
e = 0.8
36. Foci:
( 4, −3)
and ( −4, − 3)
a=7
47. Foci:
( 3, 0 )
and ( −3, 0 )
Sum of the focal radii = 8
37. Foci:
a=6
( −8, 1)
and ( 2, 1)
48. Foci:
( 0,
)
(
11 and 0, − 11
)
Sum of the focal radii = 12
116
University of Houston Department of Mathematics
Exercise Set 8.2: Ellipses
Circle is tangent to the y-axis
A circle is a special case of an ellipse where a = b . It
2
2
2
2
2
then follows that c = a − b = a − a = 0 , so c = 0 .
(Therefore, the foci are at the center of the circle, and
this is simply labeled as the center and not the focus.)
The standard form for the equation of a circle is
( x − h) 2 + ( y − k )2 = r 2
(Note: If each term in the above equation were
divided by r 2 , it would look like the standard form for
an ellipse, with a = b = r .)
Using the above information, use the given features of
each of the following circles to write an equation for
the circle in standard form.
49. Center:
( 0, 0 )
Answer the following for each circle. For answers
involving radicals, give exact answers and then round
to the nearest tenth.
(a) Write the given equation in the standard form
for the equation of a circle. (Some equations
may already be given in standard form.)
(b) State the coordinates of the center.
(c) State the length of the radius.
(d) Sketch a graph of the circle which includes
the features from (b) and (c). Label the center
C and show four points on the circle itself
(these four points are equivalent to the
vertices of the major and minor axes for an
ellipse).
Radius: 9
50. Center:
62. x 2 + y 2 − 8 = 0
5
Radius:
51. Center:
61. x 2 + y 2 − 36 = 0
( 0, 0 )
( 7, − 2 )
63.
Radius: 10
52. Center:
16
( −3, − 4 )
Radius: 3 2
54. Center:
( 2, − 5 )
16
2
=1
65.
( x + 5 ) 2 + ( y − 2 )2 = 4
66.
( x − 1)2 + ( y − 4 )2 = 36
67.
( x + 4 )2 + ( y + 3)2 = 12
68.
( x − 1)2 + ( y − 5 )2 = 7
69. x 2 + y 2 + 2 x − 10 y + 17 = 0
( −6, − 3)
70. x 2 + y 2 + 6 x − 2 y − 6 = 0
Passes through the point ( −8, − 2 )
57. Endpoints of diameter:
( −4, − 6 )
58. Endpoints of diameter:
( 3, 0 )
59. Center:
( y + 2)
x2 ( y + 2)
+
=1
9
9
( −8, 0 )
Passes through the point ( 7, − 6 )
56. Center:
+
64.
Radius: 2 5
55. Center:
2
2
( −2, 5 )
Radius: 7
53. Center:
( x − 3)
( −3, 5)
71. x 2 + y 2 + 10 x − 8 y + 36 = 0
and ( −2, 0 )
and ( 7,10 )
72. x 2 + y 2 − 4 x − 14 y + 50 = 0
73. 3x 2 + 3 y 2 + 18 x − 24 y + 63 = 0
74. 2 x 2 + 2 y 2 + 10 x + 12 y + 52 = 10
Circle is tangent to the x-axis
60. Center:
( −3, 5)
MATH 1330 Precalculus
117
Exercise Set 8.2: Ellipses
Answer the following.
75. A circle passes through the points ( 7, 6 ) ,
( 7, − 2 )
and (1, 6 ) . Write the equation of the
circle in standard form.
76. A circle passes through the points ( −4, 3) ,
( −2, 3)
and ( −2, − 1) . Write the equation of the
circle in standard form.
77. A circle passes through the points ( 2,1) ,
( 2, − 3)
and ( 8, 1) . Write the equation of the
circle in standard form.
78. A circle passes through the points ( −7, − 8 ) ,
( −7, 2 )
and ( −1, 2 ) . Write the equation of the
circle in standard form.
118
University of Houston Department of Mathematics
Exercise Set 8.3: Hyperbolas
Identify the type of conic section (parabola, ellipse,
circle, or hyperbola) represented by each of the
following equations. (In the case of a circle, identify
the conic section as a circle rather than an ellipse.) Do
NOT write the equations in standard form; these
questions can instead be answered by looking at the
signs of the quadratic terms.
2 y + x2 + 9 x = 0
2.
14 x 2 + 7 x − 12 y = −6 y 2 + 95
a2
3.
7 x − 3 y = 5 x − y + 40
4.
y2 + 9 = 9 y − x
5.
3x − 7 x + 3 y = −12 y + 13
6.
x 2 + 10 x = −2 y − y 2 + 5
7.
4 y2 + 2 x2 = 8 y − 6 x + 9
8.
8 y 2 + 24 x = 8 x 2 + 30
(a) State the point-slope equation for a line.
= 1 , what is the “rise” of
a2
b2
each slant asymptote from the center? What
is the “run” of each slant asymptote from the
center?
(d) Based on the answers to part (c), what is the
slope of each of the asymptotes for the graph
Write each of the following equations in the standard
form for the equation of a hyperbola, where the
standard form is represented by one of the following
equations:
9.
−
b2
=1
( y − k )2 ( x − h)2
−
a2
−
b2
=1
y 2 − 8x2 − 8 = 0
10. 3x 2 − 10 y 2 − 30 = 0
2
2
11. x − y − 6 x = −2 y − 3
( y − k )2 − ( x − h )2
= 1 ? (Remember that
a2
b2
there are two slant asymptotes passing
through the center of the hyperbola, one
having positive slope and one having
negative slope.)
of
a2
=1 .
( y − k )2 − ( x − h )2
2
( x − h)2 ( y − k )2
−
b2
(c) Recall that the formula for slope is
rise
represented by
. In the equation
run
2
2
( y − k )2 − ( x − h )2
(b) Substitute the center of the hyperbola,
( h, k ) into the equation from part (a).
1.
2
17. The following questions establish the formulas
for the slant asymptotes of
(e) Substitute the slopes from part (d) into the
equation from part (b) to obtain the
equations of the slant asymptotes.
18. The following questions establish the formulas
for the slant asymptotes of
( x − h )2 − ( y − k )2
a2
b2
=1 .
12. 9 x 2 − 3 y 2 = 48 y + 192
(a) State the point-slope equation for a line.
13. 7 x 2 − 5 y 2 + 14 x + 20 y − 48 = 0
(b) Substitute the center of the hyperbola,
( h, k ) into the equation from part (a).
14. 9 y 2 − 2 x 2 + 90 y + 16 x + 175 = 0
Answer the following.
15. The length of the transverse axis of a hyperbola
is __________.
16. The length of the conjugate axis of a hyperbola is
__________.
(c) Recall that the formula for slope is
rise
represented by
. In the equation
run
( x − h )2 − ( y − k )2
= 1 , what is the “rise” of
a2
b2
each slant asymptote from the center? What
is the “run” of each slant asymptote from the
center?
(d) Based on the answers to part (c), what is the
slope of each of the asymptotes for the graph
MATH 1330 Precalculus
119
Exercise Set 8.3: Hyperbolas
( x − h )2 − ( y − k )2
= 1 ? (Remember that
a2
b2
there are two slant asymptotes passing
through the center of the hyperbola, one
having positive slope and one having
negative slope.)
of
(e) Substitute the slopes from part (d) into the
equation from part (b) to obtain the
equations of the slant asymptotes.
19. In the standard form for the equation of a
hyperbola, a 2 represents (choose one):
the larger denominator
the denominator of the first term
20. In the standard form for the equation of a
hyperbola, b 2 represents (choose one):
the smaller denominator
the denominator of the second term
22.
x2 y2
−
=1
36 25
23. 9 x 2 − 25 y 2 − 225 = 0
24. 16 y 2 − x 2 − 16 = 0
25.
( x + 1)2 ( y − 5 )2
−
16
26.
( x + 5)
2
4
27.
( y − 3)
2
( y − 6)
64
( y + 2)
−
−
2
−
( x − 1)
=1
2
=1
16
25
28.
9
2
=1
36
( x + 4)
36
2
=1
29. x 2 − 25 y 2 + 8 x − 150 y − 234 = 0
30. 4 y 2 − 81x 2 = −162 x + 405
Answer the following for each hyperbola. For answers
involving radicals, give exact answers and then round
to the nearest tenth.
31. 64 x 2 − 9 y 2 + 18 y = 521 − 128 x
(a) Write the given equation in the standard form
for the equation of a hyperbola. (Some
equations may already be given in standard
form.)
33. 5 y 2 − 4 x 2 − 50 y − 24 x + 69 = 0
It may be helpful to begin sketching the graph for
part (h) as a visual aid to answer the questions
below.
35. x 2 − 3 y 2 = 18 x + 27
32. 16 x 2 − 9 y 2 − 64 x − 18 y − 89 = 0
34. 7 x 2 − 9 y 2 − 72 y = 32 − 70 x
36. 4 y 2 − 21x 2 − 8 y − 42 x − 89 = 0
(b) State the coordinates of the center.
(c) State the coordinates of the vertices, and then
state the length of the transverse axis.
(d) State the coordinates of the endpoints of the
conjugate axis, and then state the length of the
conjugate axis.
(e) State the coordinates of the foci.
(f) State the equations of the asymptotes.
(Answers may be left in point-slope form.)
(g) State the eccentricity.
(h) Sketch a graph of the hyperbola which
includes the features from (b)-(f), along with
the central rectangle. Label the center C, the
vertices V1 and V2, and the foci F1 and F2.
y2 x2
21.
−
=1
9 49
120
Use the given features of each of the following
hyperbolas to write an equation for the hyperbola in
standard form.
37. Center:
( 0, 0 )
a =8
b=5
Horizontal Transverse Axis
38. Center:
( 0, 0 )
a=7
b=3
Vertical Transverse Axis
39. Center:
( −2, − 5 )
a=2
b = 10
Vertical Transverse Axis
University of Houston Department of Mathematics
Exercise Set 8.3: Hyperbolas
40. Center:
( 3, − 4 )
52. Center:
a =1
b=6
Horizontal Transverse Axis
41. Center:
( −6, 1)
Length of transverse axis: 10
Length of conjugate axis: 8
Vertical Transverse Axis
Vertex:
Equation of one asymptote:
6 x + 5 y = −7
53. Vertices:
( 2, 5 )
Length of transverse axis: 6
Length of conjugate axis: 14
Horizontal Transverse Axis
43. Foci:
( 0, 9 )
and ( 0, − 9 )
Length of transverse axis: 6
44. Foci:
( 5, 0 )
and ( −5, 0 )
Length of conjugate axis: 4
45. Foci:
( −2, 3)
and (10, 3)
Length of conjugate axis: 10
46. Foci:
( −3, 8 )
and ( −3, − 6 )
Length of transverse axis: 8
47. Vertices:
( 4, − 7 )
and ( 4, 9 )
e=
and ( 7, 4 )
( 2, 6 )
and ( 2, − 1)
7
3
( 4, − 3)
One focus is at ( 4, 6 )
55. Center:
e=
3
2
( −1, − 2 )
One focus is at ( 9, − 2 )
56. Center:
e=
5
2
57. Foci:
e=
( 3, 0 )
and ( 3, 8 )
4
3
58. Foci:
b=4
( −1, 4 )
e=3
54. Vertices:
42. Center:
( −1, 2 )
( 3, 2 )
( −6, − 5 )
and ( −6, 7 )
e=2
48. Vertices: (1, 6 ) and ( 7, 6 )
b=7
49. Center:
( 5, 3)
One focus is at ( 5, 8)
One vertex is at ( 5, 6 )
( −3, − 4 )
One focus is at ( 7, − 4 )
One vertex is at ( 5, − 4 )
50. Center:
51. Center:
Vertex:
( −1, 2 )
( 3, 2 )
Equation of one asymptote:
7 x − 4 y = −15
MATH 1330 Precalculus
121
Exercise Set A.1: Factoring Polynomials
At times, it can be difficult to tell whether or not a
quadratic of the form ax 2 + bx + c can be factored
into the form ( dx + e ) ( fx + g ) , where a, b, c, d, e, f,
and g are integers. If b 2 − 4ac is a perfect square, then
the quadratic can be factored in the above manner.
For each of the following problems,
(a) Compute b 2 − 4ac .
(b) Use the information from part (a) to
determine whether or not the quadratic can
be written as factors with integer coefficients.
(Do not factor; simply answer Yes or No.)
1.
2
21. x 2 + 16 x + 64
22. x 2 − 6 x + 9
23. x 2 − 15 x + 56
24. x 2 − 6 x − 27
25. x 2 − 11x − 60
26. x 2 + 19 x + 48
27. x 2 + 17 x + 42
28. x 2 − 12 x − 64
x − 5x + 3
2
29. x 2 − 49
2.
x − 7 x + 10
3.
x 2 + 6 x − 16
4.
x2 + 6 x + 4
5.
9 − x2
6.
7x − x 2
7.
2 x2 − 7 x − 4
34. 16 x 2 − 81
8.
6 x2 − x −1
35. 2 x 2 − 5 x − 3
9.
2 x2 + 2 x + 5
36. 3x 2 + 16 x + 15
10. 5 x 2 − 4 x + 1
30. x 2 + 36
31. x 2 − 3
32. x 2 − 8
33. 9 x 2 + 25
37. 8 x 2 − 2 x − 3
38. 4 x 2 − 16 x + 15
Factor the following polynomials. If the polynomial
can not be rewritten as factors with integer
coefficients, then write the original polynomial as your
answer.
39. 9 x 2 + 9 x − 4
40. 5 x 2 + 17 x + 6
11. x 2 + 4 x − 5
41. 4 x 2 − 3x − 10
12. x 2 + 9 x + 14
42. 9 x 2 + 21x + 10
13. x 2 − 5 x + 6
43. 12 x 2 − 17 x + 6
14. x 2 − x − 6
44. 8 x 2 + 26 x − 7
15. x 2 − 7 x − 12
16. x 2 − 8 x + 15
17. x 2 + 12 x + 20
Factor the following. Remember to first factor out the
Greatest Common Factor (GCF) of the terms of the
polynomial, and to factor out a negative if the leading
coefficient is negative.
18. x 2 + 7 x + 18
19. x 2 − 5 x − 24
2
20. x + 9 x − 36
45. x 2 + 9 x
46. x 2 − 16 x
47. −5 x 2 + 20 x
122
University of Houston Department of Mathematics
Exercise Set A.1: Factoring Polynomials
48. 4 x 2 − 28 x
49. 2 x 2 − 18
50. −8 x 2 + 8
51. −5 x 4 + 20 x 2
52. 3x3 − 75 x
53. 2 x 2 + 10 x + 8
54. 3x 2 + 12 x − 63
55. −10 x 2 + 10 x + 420
56. −4 x 2 + 40 x − 100
57. x 3 + 9 x 2 − 22 x
58. x 3 − 7 x 2 − 6 x
59. − x3 − 4 x 2 − 4 x
60. x 5 + 10 x 4 + 21x3
61. x 4 + 6 x3 + 6 x 2
62. − x 3 − 2 x 2 + 80 x
63. 9 x5 − 100 x3
64. 49 x12 − 64 x10
65. 50 x 2 + 55 x + 15
66. 30 x 2 + 24 x − 72
Factor the following polynomials. (Hint: Factor first
by grouping, and then continue to factor if possible.)
67. x 3 + 2 x 2 − 25 x − 50
68. x 3 − 3x 2 − 4 x + 12
69. x 3 − 5 x 2 + 4 x − 20
70. 9 x3 + 18 x 2 − 25 x − 50
71. 4 x 3 + 36 x 2 − x − 9
72. 9 x3 − 27 x 2 + 4 x − 12
MATH 1330 Precalculus
123
Exercise Set A.2: Dividing Polynomials
Use long division to find the quotient and the
remainder.
Use synthetic division to find the quotient and the
remainder.
73.
x 2 − 6 x + 11
x−2
87.
x2 − 8x + 4
x − 10
74.
x 2 + 5 x + 12
x+3
88.
x2 − 4x − 6
x+3
75.
x2 + 7 x − 2
x +1
89.
3x 3 + 13 x 2 − 6 x + 28
x+5
76.
x2 − 6 x − 5
x−4
90.
2 x3 − x 2 − 31x
x−4
91.
x 4 + 3x 2 − 4
x +1
x 3 − 2 x 2 − 22 x + 33
78.
x−5
92.
2 x5 + 3 x 4 − 7 x + 8
x −1
6 x3 + 5 x 2 + 6 x − 12
79.
2x −1
93.
3 x 4 − 11x3 − 27 x 2 + 18 x + 10
x −5
12 x 3 + 13 x 2 − 22 x − 14
3x + 4
94.
2 x 4 + 3 x3 − 18 x 2 + 5 x − 12
x−2
2 x3 + 13x 2 + 28 x + 21
81.
x2 + 3x + 1
95.
x3 + 8
x+2
x 4 − 7 x 3 + 4 x 2 − 42 x − 12
82.
x2 − 7 x − 2
96.
x 4 − 81
x+3
97.
4 x3 − 7 x + 5
x − 12
98.
6 x 4 + x3 − 10 x 2 + 9
x − 13
x3 − 2 x 2 − 19 x − 12
77.
x+3
80.
83.
2 x5 + 32 x 4 + 3 x3 + 44 x 2 − 14
84.
10 x8 + 20 x 6 + x 4 + 2 x3 + 28 x 2 + 4
85.
3x 4 − x3 − 15
86.
3x 5 − 4 x3 − 2 x + 7
4x2 + 6
2x4 + 6x2 + 3
x2 + 5
x2 − 2x
Evaluate P(c) using the following two methods:
(a) Substitute c into the function.
(b) Use synthetic division along with the
Remainder Theorem.
99. P( x ) = x3 − 4 x 2 + 5 x + 2; c = 2
100. P( x) = 5 x 3 + 7 x 2 − 8 x − 3; c = −1
124
University of Houston Department of Mathematics
Exercise Set A.2: Dividing Polynomials
101. P( x ) = 7 x3 + 8 x 2 − 4 x − 12; c = −1
113.
2 x2 + 7 x + 5
x +1
114.
5 x 2 + 4 x − 12
x+2
102. P( x) = 2 x 4 − 7 x3 + 6 x − 14; c = 3
Evaluate P(c) using synthetic division along with the
Remainder Theorem. (Notice that substitution without
a calculator would be quite tedious in these examples,
so synthetic division is particularly useful.)
103. P( x ) = 3x 7 − 8 x 6 − 38 x5 + 70 x3 + 21x 2 + 3; c = 5
104. P( x ) = x 6 + 3x 5 + 10 x 4 − 35 x 2 − 2 x − 11; c = −2
105. P( x ) = 4 x 4 − 5 x3 − 22 x 2 − 1; c = − 34
106. P( x ) = 6 x 6 − 19 x5 + x 4 − 32 x 3 + 59 x + 13; c = 72
When the remainder is zero, the dividend can be
written as a product of two factors (the divisor and the
quotient), as shown below.
30
= 6 , so 30 = 5 ⋅ 6 .
5
x2 + x − 6
= x − 2 , so x 2 + x − 6 = ( x + 3 ) ( x − 2 )
x+3
In the following examples, use either long division or
synthetic division to find the quotient, and then write
the dividend as a product of two factors.
107.
x 2 − 11x + 24
x −8
108.
x 2 + 3 x − 40
x−5
109.
x 2 − 7 x − 18
x+2
110.
x 2 + 10 x + 21
x+3
111.
4 x 2 − 25 x − 21
x−7
112.
3 x 2 − 22 x + 24
x−6
MATH 1330 Precalculus
125
Odd-Numbered Answers to Exercise Set 1.1: An Introduction to Functions
1.
This mapping does not make sense, since Erik could
not record two different temperatures at 9AM.
31. (a) Domain: [ 0, ∞ )
Range:
[ 0, ∞ )
No, the mapping is not a function.
(b) Domain: [ 0, ∞ )
3.
Yes, the mapping is a function.
Domain: {−3, 7, 9}
Range:
Range:
(c) Domain: [ 6, ∞ )
{0, 4, 5}
Range:
5.
[ −6, ∞ )
[ 0, ∞ )
No, the mapping is not a function.
(d) Domain: [ 6, ∞ )
7.
x
f ( x) = + 4
7
9.
f ( x) = x − 6
Range: [3, ∞ )
Range:
11. x ≠ 3
Interval Notation: ( −∞, 3) ∪ ( 3, ∞ )
Range:
15. x ≠ 4 and x ≠ 7
Interval Notation: ( −∞, 4 ) ∪ ( 4, 7 ) ∪ ( 7, ∞ )
Range:
19. All real numbers
Interval Notation: ( −∞, ∞ )
( −∞, ∞ )
[ 2, ∞ )
(f) Domain:
Range:
21. x ≥ 5
( −∞, ∞ )
[ 0, ∞ )
35. (a) Domain:
Interval Notation: [5, ∞ )
Range:
and x ≠ −4
( −∞, ∞ )
[ 0, ∞ )
(b) Domain:
3
2
25. All real numbers
Interval Notation: ( −∞, ∞ )
27. t ≠ −5
Interval Notation: ( −∞, − 5 ) ∪ ( −5, ∞ )
29. t ≤ 4 and t ≥ 6
Interval Notation: ( −∞, 4] ∪ [ 6, ∞ )
[ −2, 2]
( −∞, 2]
(e) Domain:
Interval Notation: [ 0, ∞ )
Interval Notation: ( −∞, − 4 ) ∪ ( −4,
( −∞, − 2] ∪ [ 2, ∞ )
[ 0, ∞ )
(d) Domain:
Range:
17. t ≥ 0
( −∞, ∞ )
[ 4, ∞ )
(c) Domain:
Range:
23. x ≤
( −∞, ∞ )
( −∞, 4]
(b) Domain:
13. x ≠ 3 and x ≠ −3
Interval Notation: ( −∞, − 3) ∪ ( −3, 3) ∪ ( 3, ∞ )
3
2
( −∞, ∞ )
[ −4, ∞ )
33. (a) Domain:
Range:
( −∞, ∞ )
( −∞, 9]
(c) Domain:
Range:
( −∞, ∞ )
[3, ∞ )
37. (a) Domain:
Range:
( −∞, ∞ )
[ −1, ∞ )
(b) Domain:
Range:
( −∞, ∞ )
[11, ∞ )
(c) Domain:
Range:
126
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 1.1: An Introduction to Functions
( −∞, ∞ )
[5, ∞ )
39. Domain:
Range:
51. f (6) = 7
f (2) = −1
f (−3) = −6
( −∞, ∞ )
( −∞, ∞ )
41. Domain:
Range:
( −∞, − 2 ) ∪ ( −2, ∞ )
( −∞, 0 ) ∪ ( 0, ∞ )
f (0) = 3
f ( 4) = 3
f
43. (a) Domain:
Range:
( −∞, 2 ) ∪ ( 2, ∞ )
( −∞, 1) ∪ (1, ∞ )
(b) Domain:
Range:
(92 ) = 4
53. f (0) = 4
f (−6) = 108
f ( 2) = 7
f (1) = 4
45. f (3) = 11
f ( 4) = 9
( )
f − 12 = − 13
2
f
f ( a ) = 5a − 4
f (a + 3) = 5a + 11
(32 ) = 4
55. Yes
f (a ) + 3 = 5a − 1
f (a ) + f (3) = 5a + 7
57. Yes
59. No
47. g (0) = 4
( )=
g−
61. Yes
37
8
1
4
63. Yes
2
g ( x + 5) = x + 7 x + 14
g
(a1 ) = a1 − a3 + 4 =
2
1 − 3a + 4 a 2
a2
65. No
67. f ( x + h) = 7 x + 7h − 4
2
g (3a ) = 9a − 9a + 4
2
3 g (a ) = 3a − 9a + 12
69. f ( x + h) = x 2 + 2 xh + h 2 + 5 x + 5h − 2
49. f (−7) = 12
f ( x + h) − f ( x )
= 2x + h + 5
h
f (0) = − 23
f
(35 ) = − 1312
f (t ) =
(
71. f ( x + h) = −8
2 +t
t−3
)
f t2 − 3 =
f ( x + h) − f ( x )
=7
h
f ( x + h) − f ( x )
=0
h
t2− 1
t2 − 6
73. f ( x + h) =
1
x+h
f ( x + h) − f ( x )
−1
=
h
x ( x + h)
MATH 1330 Precalculus
127
Odd-Numbered Answers to Exercise Set 1.2: Functions and Graphs
1.
No, the graph does not represent a function.
3.
Yes, the graph represents a function.
5.
Yes, the graph represents a function.
7.
No, the graph does not represent a function.
9.
Yes, the graph represents a function.
17. (a) Domain:
(b) Range:
( −∞, 6 )
( −∞, 5]
(c) The function has two x-intercepts.
(d) g (−2) = 1
11. (a)
g (0) = 5
g (2) = −1
g (4) = 1
g (6) is undefined
y
(e) g (−2) is greater than g (3) , since 1 > 0 .
6
(f) g is increasing on the interval ( −∞, 0 ) ∪ ( 2, 6 ) .
4
(g) g is increasing on the interval ( 0, 2 ) .
2
19. (a) Domain:
x
−4
−2
2
( −∞, ∞ )
(b) x-intercept: 4;
y-intercept: 6
4
(c)
−2
(b) No, the set of points does not represent a
function. The graph does not pass the vertical
line test at x = 2 .
f ( x ) = − 32 x + 6
9
x
−2
−1
= 7.5
6
9
2
= 4.5
3
0
1
13. If each x value is paired with only one y value, then
the set of points represents a function. If an x value is
paired with more than one y value (i.e. two or more
coordinates have the same x value but different y
values), then the set of points does not represent a
function.
15
2
2
y
10
8
[ −4, 6]
[ −3, 9]
15. (a) Domain:
(b) Range:
6
4
(c) y-intercept: −3
2
(d) f (−2) = 3
f (0) = −3
f (4) = 9
f (6) = 3
x
−6
−4
−2
2
4
6
−2
(e) x = −4 , x = 4
(f) f is increasing on the interval ( 0, 4 ) .
(g) f is decreasing on the interval ( −4, 0 ) ∪ ( 4, 6 ) .
(h) The maximum value of the function is 9 .
21. (a) Domain:
[ −1, 3)
(b) x-intercept:
5
; y-intercept: −5
3
(i) The minimum value of the function is −3 .
128
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 1.2: Functions and Graphs
21. (c)
x
−1
h( x ) = 3 x − 5
−8
0
−5
1
2
3
25. (a) Domain: [3, ∞ )
(b) x-intercept: 3;
(c)
y-intercept: None
−2
x
3
f ( x) = x − 3
0
1
4
1
undefined
5
(open circle at y = 4 )
y
2 ≈ 1.4
7
2
12
3
4
2
y
8
x
−6
−4
−2
2
4
6
8
6
−2
4
−4
2
−6
x
−2
−8
2
4
6
8
10
12
−2
−10
−4
(b) x-intercept: 3;
(c)
−6
( −∞, ∞ )
23. (a) Domain:
y-intercept: 3
x
1
g( x ) = x − 3
2
2
1
3
27. (a) Domain:
( −∞, ∞ )
(b) x-intercepts: 0, 4
(c)
y-intercept: 0
x
−1
F ( x) = x 2 − 4 x
5
0
0
0
4
1
1
−3
5
2
2
−4
3
−3
y
6
4
y
4
2
2
x
−2
2
4
6
x
8
−4
−2
−2
2
4
6
8
−2
−4
−6
MATH 1330 Precalculus
129
Odd-Numbered Answers to Exercise Set 1.2: Functions and Graphs
29. (a) Domain: (− ∞, ∞ )
(b) x-intercept: −1 ;
(c)
31. (c) Graph:
y-intercept: 1
x
−2
f ( x ) = x3 + 1
−7
−1
0
0
1
1
2
2
9
y
8
6
4
33. (a) Domain:
2
x
−8
−6
−4
−2
2
4
6
8
−2
−4
−6
[ −2, 5]
(b) y-intercept: 4
(c)
x
−2
f ( x)
0
−1
2
0
4
2
1
31. (a) Domain: (− ∞, 0) ∪ (0, ∞ )
(b) x-intercepts: None; y-intercept: None
(c)
x
−6
g( x ) = 12x
−2
−4
−3
−3
−4
−2
−6
2
6
3
4
4
3
6
2
(open circle at y = 6 )
2
1
3
0
4
−1
5
−2
y
6
4
2
x
−4
−2
2
4
6
−2
130
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 1.2: Functions and Graphs
35. (a) Domain:
( −∞, ∞ )
39. (a) Domain:
(b) y-intercept: 0
(b) y-intercept: −5
(c)
( −∞, ∞ )
(c)
x
−6
f ( x)
3
x
−6
−2
−5
−3
f ( x)
−6
−3
(open circle at y = 9 )
(open circle at y = 3 )
−5
2
y
−2
4
−1
1
0
0
1
1
2
4
5
4
4
2
x
−8
−6
−4
−2
2
4
−2
10
y
8
−4
6
−6
4
2
x
−8
37. (a) Domain:
( −∞, ∞ )
−4
−2
2
4
6
8
10
−2
(b) y-intercept: 1
(c)
−6
−4
−6
x
−3
f ( x)
4
0
41. (a) origin
(b) y-axis
1
(open circle at y = 3 )
1
2
2
5
43.
( 3, − 6 )
45. Odd
6
47. Even
y
49. Neither
4
2
x
−4
−2
2
4
−2
MATH 1330 Precalculus
131
Odd-Numbered Answers to Exercise Set 1.3: Transformations of Graphs
1.
(a)
(b)
(c)
(d)
17. E
Shift right 3 units.
Reflect in the y-axis.
Stretch vertically by a factor of 2.
Shift upward 4 units.
19. Note: When two equations are given, either is
acceptable.
3
f ( x ) = −5 ( x − 7 ) + 1
(a)
3.
(a) Jack is correct.
(b) Analysis of Jack’s Method:
Base graph
y = x2
3
(b) f ( x ) = −5 ( x − 7 ) + 1
f ( x ) = − 5 ( x − 7 ) + 1 = −5 ( x − 7 ) − 1
3
3
(d) f ( x ) = −5 ( x − 7 ) + 1 = −5 ( x − 7 ) − 5
3
(c)
Reflect in the x-axis
y = − x2
Shift upward 2 units
y = − x 2 + 2 = 2 − x2
3
f ( x ) = −5 ( x + 7 ) + 1
(e)
Analysis of Jill’s Method:
Base graph
y = x2
Shift upward 2 units
Reflect in the x-axis
(f) The equations in parts (a) and (b) are equivalent.
y = x2 + 2
(
)
(a) Tony is correct.
(b) Analysis of Tony’s Method:
Base graph
y= x
Shift right 2 units
Reflect in the y-axis
y = x−2
y = − x − 2 = −2 − x
Analysis of Maria’s Method:
Base graph
y= x
Reflect in the y-axis
Shift right 2 units
21. Reflect in the y-axis, then shift downward 2 units.
y = − x 2 + 2 = −2 − x 2
Jack’s method is correct, as the final result
matches the given equation, y = 2 − x 2 . Jill’s
method, however, is incorrect, as it yields the
equation y = −2 − x 2 .
5.
3
y = −x
23. Shift right 2 units, then stretch vertically by a factor
of 5, then reflect in the x-axis, then shift upward 1
unit.
25. Shift left 8 units, then stretch vertically by a factor of
3, then shift upward 2 units.
27. Shift upward 1 unit.
29. Reflect in the y-axis, then shift upward 3 units.
31. Shift right 2 units, then shrink vertically by a factor
of 14 , then reflect in the x-axis, then shift downward
5 units.
33. Shift left 7 units, then reflect in the y-axis, then shift
upward 2 units.
y
35.
y = − ( x − 2) = 2 − x
2
x
Tony’s method is correct, as the final result
matches the given equation, y = −2 − x .
Maria’s method, however, is incorrect, as it
−4
−2
9.
D
4
6
−2
−4
yields the equation y = − ( x − 2 ) = 2 − x .
7.
2
−6
37.
y
4
F
2
11. A
x
−6
−4
−2
2
13. K
−2
15. C
−4
132
4
6
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 1.3: Transformations of Graphs
y
39.
49.
10
8 y
6
8
4
6
2
4
x
2
−10
−8
−6
−4
−2
x
−6
−4
−2
2
41.
4
2
−2
6
−4
y
51.
y
4
6
2
4
x
−2
2
x
−4
−2
2
2
−4
y
53.
4
y
−2
Notice the
asymptotes
(the blue
dotted lines)
at x = 3 and
10
8
2
−4
6
4
−2
43.
4
−2
2
4
6
x
6
8
4
−2
y = 6.
2
x
−4
−2
−6
2
4
6
8
10
−8
55.
y
Notice the
asymptotes
(the blue
dotted lines)
at x = 0 and
8
45.
6
8 y
4
6
2
x
4
−6
−4
−2
2
4
y = 3.
6
−2
2
x
−2
2
4
6
y
57.
8
6
4
47.
8 y
2
6
−8 −6 −4 −2
x
2
4
6
8
−2
4
−4
−6
2
x
−4
−2
2
4
6
−2
MATH 1330 Precalculus
59. All of the points with negative y-values have
reflected over the x-axis so that they have positive yvalues.
133
Odd-Numbered Answers to Exercise Set 1.3: Transformations of Graphs
61. (a)
67.
y
10
8
6
y
8
4
2
x
−10 −8 −6 −4 −2
−2
2
4
6
8
6
10
4
−4
−6
2
−8
x
−10
−8
−6
−4
−2
2
4
6
8
10
−2
(b)
y
8
−4
6
4
−6
2
−10 −8 −6 −4 −2
−2
x
2
4
6
8
−8
10
−4
−6
−8
−10
69.
63.
10
10
8
y
6
8
4
6
2
4
x
2
−8
−6
−4
−2
x
−8
−6
−4
y
−2
2
4
6
8
2
4
6
8
10
−2
10
−4
−2
−6
−4
−8
−6
−8
71.
65.
10
10
y
y
8
8
6
6
4
4
2
x
2
x
−8
−6
−4
−2
2
−2
−4
−6
4
6
8
10
−8
−6
−4
−2
2
4
6
8
10
−2
−4
−6
−8
−8
134
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 1.4: Operations on Functions
1.
(a) f (−3) + g (−3) = 4 + 0 = 4 .
(c) ( fg )( x) =
(b) f (0) + g (0) = 2 + 3 = 5 .
(c)
3x
( x − 1)( x + 2)
Domain:
f (−6) + g (−6) is undefined, since g (−6) is
i.e. x ≠ −2 and x ≠ 1 .
undefined.
(d) f (5) + g (5) = 0 + 6 = 6 .
f
3x + 6
( x) =
g
x
( x − 1)
(d)
(e)
f (7) + g (7) is undefined, since f (7) is
undefined.
(f) Graph of f + g :
8
6
(− ∞, − 2) ∪ (−2, 1) ∪ (1, ∞),
Domain:
(− ∞, − 2) ∪ (−2, 0) ∪ (0, 1) ∪ (1, ∞),
i.e. x ≠ −2 and x ≠ 0 and x ≠ 1 .
y
f+g
7.
4
(a) ( f + g )( x ) = x − 6 + 10 − x
Domain: [6, 10]
2
(b) ( f − g )( x) = x − 6 − 10 − x
Domain: [6, 10]
x
−8
−6
−4
−2
2
4
6
8
−2
(c) ( fg )( x) = x − 6 ⋅ 10 − x = − x 2 + 16 x − 60
Domain: [6, 10]
−4
−6
−8
f
g
(d)
(g) The domain of f + g is [− 3, 6] , since this is
where the domains of f and g intersect (overlap).
3.
(a) ( f + g )( x ) = x 2 − 2 x − 9
Domain: (− ∞, ∞ )
(b) ( f − g )( x) = − x 2 + 6 x + 15
Domain: (− ∞, ∞ )
(c) ( fg )( x) = 2 x3 − 5 x 2 − 36 x − 36
Domain: (− ∞, ∞ )
f
2x + 3
2x + 3
( x) =
=
2
g
(
x
+
2)( x − 6)
x − 4 x + 12
(d)
Domain:
Domain: [6, 10 ) .
9.
(a) ( f + g )( x ) = x 2 − 9 + x 2 + 4
Domain: (− ∞, − 3] ∪ [3, ∞ )
(b) ( f − g )( x) = x 2 − 9 − x 2 + 4
Domain: (− ∞, − 3] ∪ [3, ∞ )
(c) ( fg )( x) = x 2 − 9 ⋅ x 2 + 4 = x 4 − 5 x 2 − 36
Domain: (− ∞, − 3] ∪ [3, ∞ )
f
g
(d)
(a) ( f + g )( x) =
Domain:
x2 + 2x + 6
( x − 1)( x + 2)
Domain:
x2 − 9
=
x2 + 4
x2 − 9
x2 + 4
(− ∞, − 3] ∪ [3, ∞ ) .
11. [1, 3) ∪ (3, ∞ )
13. (− ∞, 2 ) ∪ (2, 7) ∪ (7, ∞)
(− ∞, − 2) ∪ (−2, 1) ∪ (1, ∞),
i.e. x ≠ −2 and x ≠ 1 .
(b) ( f − g )( x) =
Domain:
( x) =
(− ∞, − 2) ∪ (−2, 6) ∪ (6, ∞),
i.e. x ≠ −2 and x ≠ 6 .
5.
x−6
x−6
( x) =
=
10
−x
10
−
x
− x2 + 4x + 6
( x − 1)( x + 2)
(− ∞, − 2) ∪ (−2, 1) ∪ (1, ∞),
i.e. x ≠ −2 and x ≠ 1 .
MATH 1330 Precalculus
15. [− 2, 5) ∪ (5, ∞ )
17. (a) g (2) = 5
(c) f (2) = 0
19. (a)
( f g )(− 3) = 2
(b) f (g (2)) = 0
(d) g ( f (2)) = 3
(b)
(g f )(− 3) = 6
135
Odd-Numbered Answers to Exercise Set 1.4: Operations on Functions
21. (a)
( f f )(3) = 3
23. (a)
(b)
( f g )(4) is undefined, since f (6) is undefined.
(g f )(4) = 1
(b)
(g g )(− 2) = 4
43. (a) f (g (1)) = 11
(b) g ( f (1)) = 7
(c)
(d) g ( f (x )) = 5 x 2 + 2
(e) f ( f (1)) = 11
(f) g (g (1)) = −23
(b) f (g (0 )) = −5
(d) g ( f (0 )) = −2
25. (a) g (0) = 4
(c) f (0) = 3
27. (a)
( f g )(− 2) = −33 (b) (g f )(− 2) = 18
29. (a)
( f f )(6) = 21
(b)
(g g )(6) = 54
f (g (x )) = 25 x 2 − 80 x + 66
(g) f ( f (x )) = x 4 + 4 x 2 + 6
(h) g (g (x )) = 25 x − 48
45. (a) f (g (− 2 )) = − 174
31. (a)
(b) g ( f (− 2 )) = − 12
19
33. (a) { x | x ≥ 5 }
x−2
13 − 2 x
3x − 6
(d) g ( f (x )) =
11 − 4 x
( f g )(x ) = −2 x 2 + 10 x − 5
(b) (g f )(x ) = 4 x 2 − 2 x − 2
(b) ( f g )( x ) =
1
x−5
(c) { x | x ≠ 5 }
(d) { x | x > 5 } ; Interval Notation: ( 5, ∞ )
(c)
f (g (x )) =
47. (a) f (g (h(2 ))) = 195
(b) g (h( f (3))) = −50
( f g h )(x ) = 36 x 2 + 24 x + 3
(d) (g h f )(x ) = 4 − 6 x 2
(c)
35. (a) { x | x ≥ 6 }
(b) ( f g )( x ) =
3
x − 10
Interval Notation: [6, 10 ) ∪ (10, ∞ )
49. (a) h( f (g (4 ))) = 23
(b) f (g (h(0 ))) = 8
(c) ( f g h )(x ) = 2 x + 8
(d) (h f g )(x ) = 2 x + 15
( f g )(x ) = 4 x 2 − 22 x + 28
51.
(c) { x | x ≠ 10 }
(d) { x | x ≥ 6 and x ≠ 10 } ;
37. (a)
Domain of f g : (−∞, ∞)
(b)
(g f )(x ) = 2 x
2
x
f ( g ( x ))
0
1
2
4
7
undefined
2
4
x
0
1
2
4
f ( f ( x ))
4
7
7
undefined
+ 6x − 7
Domain of g f : (−∞, ∞)
53.
1
39. (a) ( f g )(x ) =
x−4
Domain of f g : (4, ∞)
(b)
(g f )(x ) =
1
2
x −4
Domain of g f : (−∞, − 2) ∪ (2, ∞)
41. (a)
(b)
136
( f g )(x ) =
2− x
Domain of f g : (− ∞, 2]
(g f )(x ) = −5 −
x+7
Domain of g f : [− 7, ∞ )
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 1.5: Inverse Functions
1.
3.
No, it is not a one-to-one function.
Explanation: The graph passes the vertical line test,
so it represents a function; but it does not pass the
horizontal line test, so it is does not represent a oneto-one function.
No, it is not a one-to-one function.
Explanation: The graph does not pass the vertical
line test, so it does not represent a function; therefore,
it cannot be a one-to-one function.
19. (c) Domain of f −1 :
Range of f
−1
:
( −∞, ∞ )
( −∞, ∞ )
21. (a) Domain of f : [ −2, 9 )
Range of f : [1, 8 )
10 y
(b)
8
f -1
6
5.
Yes, it is a one-to-one function.
Explanation: The graph passes the vertical line test,
so it represents a function; it also passes the
horizontal line test, so it represents a one-to-one
function.
7.
Yes, the function is one-to-one.
9.
Yes, the function is one-to-one.
4
2
x
−2
2
4
6
8
10
−2
(c) Domain of f −1 : [1, 8 )
Range of f −1 :
[ −2, 9 )
11. No, the function is not one-to-one.
13. No, the function is not one-to-one.
15. (a) Interchange the x and y values.
(b) Reflect the graph of f over the line y = x.
23. 4
25. −3
27. 3
29. 2
17.
x
−1
f
−4
( x)
3
31. 4
7
2
33. 3
5
−4
0
5
3
0
( −∞, ∞ )
( −∞, ∞ )
19. (a) Domain of f :
Range of f :
(Extraneous information is given in this problem.
Note that for any inverse functions f and g,
f g ( x ) = x and g f ( x ) = x. )
35. f −1 ( x ) =
x+3
5
37. f −1 ( x ) =
8x − 3 3 − 8x
=
−2
2
39. f −1 ( x ) = x − 1, where x ≥ 1
(b)
6 y
4
f
-1
x+7
4
41. f −1 ( x ) =
3
43. f −1 ( x ) =
3 − 2x
x
2
x
−6
−4
−2
2
−2
4
−4
−6
MATH 1330 Precalculus
137
Odd-Numbered Answers to Exercise Set 1.5: Inverse Functions
45. f −1 ( x ) =
4x + 3
x−2
47. f −1 ( x ) =
x2 − 7 7 − x2
=
−2
2
49. No, f and g are not inverses of each other, since
f g ( x ) ≠ x and g f ( x ) ≠ x .
51. Yes, f and g are inverses of each other, since
f g ( x ) = x and g f ( x ) = x .
53. Yes, f and g are inverses of each other, since
f g ( x ) = x and g f ( x ) = x .
55. Yes, f and g are inverses of each other, since
f g ( x ) = x and g f ( x ) = x .
57. f −1 (500) represents the number of tickets sold when
the revenue is $500.
59. Yes, f is one-to-one.
61. Yes, f is one-to-one.
63. No, f is not one-to-one.
Using x1 = 3 and x2 = −3 , for example, it can be
shown that f ( x 1 ) = f ( x2 ) . However, x1 ≠ x2 , and
therefore f is not one-to-one. (Answers vary for this
counterexample.)
65. No, f is not one-to-one.
Using x1 = 2 and x2 = −2 for example, it can be
shown that f ( x 1 ) = f ( x2 ) . However, x1 ≠ x2 , and
therefore f is not one-to-one. (Answers vary for this
counterexample.)
138
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 2.1: Linear and Quadratic Functions
5
4
1.
−
3.
0
(b)
3
4
(c)
5.
7.
9.
(d)
(e)
Undefined
f ( x) =
4
x+3
3
4
f ( x) = x + 3
3
4
Slope:
3
y-intercept: 3
Graph:
15. (a) y =
8
4
x−2
3
y
6
11. (a) y = −2 x + 5
4
(b) f ( x ) = −2 x + 5
2
(c) Slope: −2
(d) y-intercept: 5
(e) Graph:
x
−4
−2
y
2
4
−2
6
4
4
17. f ( x ) = − x + 3
7
2
2
4
19. f ( x ) = − x +
9
3
x
−2
2
4
6
21. f ( x ) = 2 x + 7
−2
23. f ( x ) =
1
13. (a) y = − x
4
1
(b) f ( x ) = − x
4
1
(c) Slope: −
4
(d) y-intercept: 0
(e) Graph:
5
x −5
7
3
25. f ( x ) = − x + 6
2
27. f ( x ) = 5
29. f ( x ) = −5 x + 18
4
y
31. f ( x ) =
2
5
x−2
2
x
−4
−2
2
4
6
2
19
33. f ( x ) = − x +
5
5
−2
35. f ( x ) = −6 x + 18
−4
−6
MATH 1330 Precalculus
37. f ( x ) = x + 8
139
Odd-Numbered Answers to Exercise Set 2.1: Linear and Quadratic Functions
1
39. f ( x ) = x − 5
3
(d) Minimum value: 3
(e) Axis of symmetry: x = 2
8
41. f ( x ) = x − 8
3
51. (a) f ( x) = −( x + 4)2 + 7
43. C ( x ) = 800 x + 6200
(b) Vertex: (−4, 7)
45. (a) f ( x) = ( x + 3)2 − 2
(c)
y
6
(b) Vertex: (−3, − 2)
(c)
4
y
4
2
x
2
−8
x
−8
−6
−4
−2
−6
−4
−2
2
−2
2
−2
(d) Maximum value: 7
−4
(e) Axis of symmetry: x = − 4
(d) Minimum value: −2
(e) Axis of symmetry: x = −3
53. (a) f ( x) = −4( x − 3)2 + 9
(b) Vertex: (3, 9)
47. (a) f ( x) = ( x − 1)2 − 1
(c)
y
10
(b) Vertex: (1, − 1)
8
6
(c)
y
6
4
2
4
x
−4
−2
2
4
6
8
10
−2
x
−4
2
−2
2
4
6
−4
(d) Maximum value: 9
−2
(e) Axis of symmetry: x = 3
(d) Minimum value: −1
(e) Axis of symmetry: x = 1
(
)2
55. (a) f ( x) = x − 52 − 134
(52 , − 134 )
(b) Vertex:
49. (a) f ( x) = 2( x − 2) 2 + 3
y
(c)
(b) Vertex: ( 2, 3)
4
(c)
2
y
8
x
−2
6
2
4
6
−2
4
−4
2
x
−4
140
−2
2
4
6
(d) Minimum value: − 134
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 2.1: Linear and Quadratic Functions
(e) Axis of symmetry: x =
67. (a) x-intercepts: −6, 2
5
2
(b) y-intercept: −36
(b) Vertex:
( −2, − 48 )
(c) Vertex:
( )2
(− 83 , 1641 )
41
57. (a) f ( x) = −4 x + 83 + 16
(d) Minimum Value: −48
(e) Axis of symmetry: x = −2
(f)
(c)
18 y
3 y
12
6
2
x
−12
−10
−8
−6
−4
−2
4
6
8
10
−6
1
−12
x
−2
2
−1
1
−18
2
−24
−1
−30
−36
41
16
(d) Maximum value:
−42
−48
(e) Axis of symmetry: x = − 83
−54
−60
59. (a) Vertex: (6, 14)
(b) MinimumValue: 14
69. (a) x-intercepts:
61. (a) Vertex: (4, 23)
(b) MaximumValue: 23
(94 , 1058 )
(b) MaximumValue:
− 52
(b) y-intercept: 5
(c) Vertex:
63. (a) Vertex:
1,
2
( −1, 9 )
(d) Maximum Value: 9
(e) Axis of symmetry: x = −1
105
8
(f)
y
10
65. (a) x-intercepts: −3, 5
(b) y-intercept: −15
(c) Vertex: (1, − 16 )
8
6
(d) Minimum value: −16
4
(e) Axis of symmetry: x = 1
2
(f)
2
y
x
x
−10
−8
−6
−4
−2
2
4
6
8
10
−8
−6
−4
−2
2
4
6
−2
−2
−4
−6
−8
−10
−12
−14
−16
−18
MATH 1330 Precalculus
141
Odd-Numbered Answers to Exercise Set 2.1: Linear and Quadratic Functions
75. (a) x-intercepts:
71. (a) x-intercepts: 3 ± 6 (approx. 0.55 and 5.45)
(b) y-intercept: 3
− 32
(b) y-intercept: 9
( 3, − 6 )
(c) Vertex:
3,
2
(c) Vertex:
(d) Minimum Value: −6
( 0, 9 )
(e) Axis of symmetry: x = 3
(d) Maximum Value: 9
(e) Axis of symmetry: x = 0
(f)
(f)
10
y
y
10
8
8
6
6
4
2
4
x
−3
−2
−1
1
2
3
4
5
6
7
8
2
−2
x
−8
−4
−6
−4
−2
2
4
6
8
−2
−6
−8
−4
2
77. f ( x ) = 3 ( x − 2 ) − 5
73. (a) x-intercepts: None
(b) y-intercept: 5
79. f ( x ) = −
(c) Vertex: (1, 4 )
3
2
( x − 5) + 7
4
(d) Minimum Value: 4
(e) Axis of symmetry: x = −1
81. The largest possible product is 25.
(f)
83. (a) 2.5 seconds
(b) 100 feet
y
10
8
6
4
2
x
−6
−4
−2
2
4
6
8
−2
142
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 2.2: Polynomial Functions
1.
(a) Yes
(b) Degree: 3
(c) Binomial
8 y
25.
6
4
2
3.
(a) Yes
(b) Degree: 1
(c) Binomial
x
−8 −6 −4 −2
−2
2
4
6
8
2
4
6
8
−4
−6
5.
(a) No
(b) N/A
(c) N/A
−8
27.
8 y
6
7.
9.
(a) Yes
(b) Degree: 2
(c) Trinomial
4
2
(a) Yes
(b) Degree: 4
(c) Trinomial
11. (a) No
(b) N/A
(c) N/A
−4
−6
−8
29.
y
17. (a) No
(b) N/A
(c) N/A
Note: The scale on the
y-axis will vary,
depending on
the value of n.
x
13. (a) No
(b) N/A
(c) N/A
15. (a) Yes
(b) Degree: 7
(c) Binomial
x
−8 −6 −4 −2
−2
−8 −6 −4 −2
0
2
4
6
8
31. (a) y = x3
(b) y = x 2
(c) x 6
19. (a) Yes
(b) Degree: 9
(c) Trinomial
33. E
21. (a)
(b)
(c)
(d)
False
True
True
False
37. F
23. (a)
(b)
(c)
(d)
True
False
False
True
35. B
39. (a) x-intercepts: 5, − 3
(b)
y-intercept: −15
y
5
x
−6
−4
−2
2
4
6
8
−5
−10
−15
MATH 1330 Precalculus
143
Odd-Numbered Answers to Exercise Set 2.2: Polynomial Functions
41. (a) x-intercept: 6
(b)
y-intercept: −36
12 y
49. (a) x-intercepts:
y-intercept: 40
(b)
x
−2
2 , − 4, 5, − 1
3
2
4
6
8
80 y
40
10 12 14
−12
−4
−24
−36
−48
43. (a) x-intercepts: 5, − 2, − 6
y-intercept: −60
x
−2 −40
−80
−120
−160
−200
−240
−280
−320
−360
−400
−440
2
6
51. (a) x-intercepts: 0, − 2, 4, − 6
(b)
(b)
4
y-intercept: 0
y
y
30
100
x
−8 −6 −4 −2
2
4
6
8
50
−30
x
−60
−6
−4
−2
2
4
6
−90
−50
−120
45. (a) x-intercepts: 4, 1, − 3
(b)
y-intercept: −6
53. (a) x-intercepts: 3, − 4
(b)
9 y
y-intercept: 144
y
144
6
3
x
−6
−4
−2
2
4
6
72
8
−3
−6
x
−9
−6
−12
−4
−2
2
4
−15
47. (a) x-intercepts: −2, 4
(b)
8
y-intercept: −16
55. (a) x-intercepts: −5, 4
(b)
y
y
x
−4
−2
2
−8
−16
−24
4
y-intercept: −500
250
6
x
−8
−6
−4
−2
2
4
6
−250
−500
−32
−750
−40
144
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 2.2: Polynomial Functions
57. (a) x-intercepts: −3, 4
(b)
y-intercept: −576
65. (a) x-intercepts: 0, − 5, 5
(b)
y
288
y
60
40
x
−6
−4
y-intercept: 0
−2
2
4
6
20
8
x
−6
−288
−4
−2
−20
2
4
6
−40
−576
−60
−80
59. (a) x-intercepts: 0, 2, − 3, 4
(b)
y-intercept: 0
67. (a) x-intercepts: 0, 4, − 3
(b)
y
y-intercept: 0
y
60
100
40
x
−4
−2
2
4
20
x
6
−4
−100
−2
2
4
6
−20
−40
−200
−60
−300
−80
61. (a) x-intercepts: 0, 1, − 1
(b)
y-intercept: 0
69. (a) x-intercepts: 0, − 3, 3
(b)
y
y-intercept: 0
y
60
0.002
40
20
x
0.001
−4
−2
x
−1
2
4
−20
1
−40
−60
−0.001
−80
63. (a) x-intercepts: 0, 2, 4
(b)
y-intercept: 0
71. (a) x-intercepts: −4, 1, − 1
(b)
y
y-intercept: −4
y
4
8
2
4
x
2
4
−2
6
x
−4
−2
2
−4
−4
−8
−6
MATH 1330 Precalculus
145
Odd-Numbered Answers to Exercise Set 2.2: Polynomial Functions
73. (a) x-intercepts: 3, − 3, 2, − 2
(b)
y-intercept: 36
y
45
36
27
18
9
x
−4
−2
2
4
−9
−18
75. P ( x ) = −
1
( x + 4 )( x − 1)( x − 3)
2
2
77. P ( x ) = 3 ( x + 2 ) ( x − 1)
79.
2
y
10
5
x
−4
−2
2
4
−5
81.
10 y
8
6
4
2
x
−4
−2
2
4
6
8
−2
83.
2 y
1
x
−3
−2
−1
1
2
3
−1
−2
−3
−4
85.
2 y
x
−6
−4
−2
2
4
−2
−4
−6
−8
146
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 2.3: Rational Functions
1.
Odd
(f) Horizontal Asymptote:
y=0
3.
Even
(g) Slant Asymptote(s):
y
(h) Graph:
6
None
5.
Neither
7.
(a)
4
2
6
y
x
−2
2
4
6
8
10
12
−2
4
−4
2
−6
x
−6
−4
−2
2
4
6
−2
13. (a) Domain:
−4
−6
(b)
( −∞, 0 ) ∪ (0, ∞)
(b) Hole(s):
None
(c) x-intercept:
− 32
(d) y-intercept(s):
None
(e) Vertical Asymptote: x = 0
y
6
4
2
(f) Horizontal Asymptote:
y=2
(g) Slant Asymptote(s):
(h) Graph:
8
None
6
x
−6
−4
−2
2
4
y
6
4
2
−2
x
−6
−4
−2
−4
2
4
6
−2
−4
−6
9.
(a) Hole:
when x = −8
(b) x-intercept:
(c) y-intercept:
15. (a) Domain:
4
−6
(d) Vertical Asymptote:
x = −2
(e) Horizontal Asymptote:
y=3
( −∞, − 3) ∪ (−3, ∞)
(b) Hole(s):
None
(c) x-intercept: 6
(d) y-intercept:
−2
(e) Vertical Asymptote:
x = −3
(f) Horizontal Asymptote:
11. (a) Domain:
( −∞, 5 ) ∪ (5, ∞)
(b) Hole(s):
None
(c) x-intercept(s):
None
(d) y-intercept:
− 35
(e) Vertical Asymptote:
(g) Slant Asymptote(s):
(h) Graph:
y =1
None
y
8
6
4
2
x=5
−10 −8 −6 −4 −2
−2
x
2
4
6
8
−4
Continued in the next column…
−6
−8
MATH 1330 Precalculus
147
Odd-Numbered Answers to Exercise Set 2.3: Rational Functions
17. (a) Domain:
(g) Slant Asymptote(s):
None
(Note: There is no slant asymptote when the
remainder after performing long division is zero.
The graph of this function is simply a line with a
removable discontinuity / hole at x = 4 .)
( −∞, − 32 ) ∪ (− 32 , ∞)
(b) Hole(s):
(c) x-intercept:
2
(d) y-intercept:
8
3
None
(h) Graph:
x = − 32
(e) Vertical Asymptote:
y
(f) Horizontal Asymptote:
y = −2
(g) Slant Asymptote(s):
(h) Graph:
None
8
6
4
y
4
2
x
2
x
−6
−4
−2
2
4
−6
−4
−2
6
2
4
−2
−2
−4
−6
23. (a) Domain:
( −∞, − 1) ∪ (−1, 1) ∪ (1, ∞)
−8
19. (a) Domain:
(b) Hole:
( −∞, 2 ) ∪ (2, 4) ∪ (4, ∞)
when x = 2
(c) x-intercept:
−3
(d) y-intercept:
− 34
(b)
(c)
(d)
(e)
Hole(s):
None
x-intercept: 0
y-intercept: 0
Vertical Asymptotes:
x = −1, x = 1
(f) Horizontal Asymptote(s): None
(g) Slant Asymptote:
y = 4x
(e) Vertical Asymptote:
(h) Graph:
x=4
(f) Horizontal Asymptote:
y =1
(g) Slant Asymptote(s):
(h) Graph:
y
None
y
16
12
8
6
4
4
x
−4
2
−2
x
−4
−2
2
4
6
8
2
4
−4
10
−2
−8
−4
−12
−6
−16
−20
21. (a) Domain:
(b) Hole:
( −∞, 4 ) ∪ (4, ∞)
when x = 4
(c) x-intercept:
−5
(d) y-intercept:
5
(e) Vertical Asymptote(s):
None
(f) Horizontal Asymptote(s): None
25. (a) Domain:
(b) Hole:
( −∞, 0 ) ∪ (0, 2) ∪ (2, ∞)
when x = 2
(c) x-intercepts: − 53
(d) y-intercepts: None
(e) Vertical Asymptote:
x=0
Continued in the next column…
Continued on the next page…
148
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 2.3: Rational Functions
(f) Horizontal Asymptote:
y = −3
(g) Slant Asymptote(s):
None
(h) Graph:
y
6
4
(h) Graph:
y
2
2
x
−6
−4
−2
2
4
6
x
−2
−6
−4
−2
2
4
6
−4
−2
−6
−4
−8
−6
27. (a) Domain:
(b) Hole:
( −∞, − 3) ∪ (−3, − 1) ∪ (−1, ∞)
(c) x-intercept:
3
(d) y-intercept:
−6
( −∞, − 2 ) ∪ (−2, 2) ∪ (2, ∞)
31. (a) Domain:
when x = −3
(e) Vertical Asymptote:
x = −1
(f) Horizontal Asymptote:
y=2
(g) Slant Asymptote(s):
(h) Graph:
None
(b)
(c)
(d)
(e)
y
6
4
Hole(s):
None
x-intercept(s):
None
−2
y-intercept:
Vertical Asymptotes:
x = −2, x = 2
(f) Horizontal Asymptotes:
y=0
(g) Slant Asymptote(s):
(h) Graph:
None
y
4
2
2
x
x
−8
−6
−4
−2
2
4
6
−6
−2
−4
−2
2
4
6
−2
−4
−4
−6
−6
29. (a) Domain:
( −∞, 0 ) ∪ (0, ∞)
33. (a) Domain:
( −∞, − 3) ∪ (−3, 4) ∪ (4, ∞)
(b) Hole(s):
None
(c) x-intercept(s):
−2, 2
(b) Hole(s):
None
(c) x-intercept: 1
(d) y-intercept(s):
None
(e) Vertical Asymptote:
(d) y-intercept:
x=0
(f) Horizontal Asymptote:
None
(g) Slant Asymptote(s):
y = − 12 x
Continued in the next column…
1
2
(e) Vertical Asymptotes:
x = −3, x = 4
(f) Horizontal Asymptote:
y=0
(g) Slant Asymptote(s):
(h) Graph:
None
y
6
4
2
x
−8
−6
−4
−2
2
4
6
8
−2
−4
−6
MATH 1330 Precalculus
149
Odd-Numbered Answers to Exercise Set 2.3: Rational Functions
35. (a) Domain:
(b) Hole:
( −∞, − 1) ∪ (1, 2) ∪ (2, 4) ∪ (4, ∞)
(h) Graph:
8
when x = 4
6
(c) x-intercepts: −2, 3
(d) y-intercept:
y
4
3
2
(e) Vertical Asymptotes:
x = −1, x = 2
x
−8
(f) Horizontal Asymptote:
y =1
(g) Slant Asymptote(s):
y
(h) Graph:
None
−6
−4
−2
2
4
6
8
−2
−4
6
−6
4
−8
2
x
−4
−2
2
4
5
13
±
;
2
2
x ≈ 0.7 ; x ≈ 4.3
6
41. (a) x-intercepts: x =
−2
−4
(b) Vertical Asymptotes:
37. (a) Domain:
(b) Holes:
when x = −1 and x = 3
(c) x-intercept:
0, 5
(d) y-intercept:
0
(e)
(f)
(g)
(h)
1
10
1
10
x=− −
; x=− +
3
3
3
3
x ≈ −1.4; x ≈ 0.7
( −∞, − 1) ∪ (−1, 3) ∪ (3, ∞)
43. (a) y = 1
(b) x = −2
(c) ( −2, 1)
Vertical Asymptote(s):
None
Horizontal Asymptote(s): None
Slant Asymptote(s):
None
Graph:
y
4
2
x
−4
−2
2
4
6
8
1
2
7
(b) x =
2
7 1
(c) ,
2 2
45. (a) y =
6
10
−2
−4
−6
39. (a) Domain:
( −∞, 0 ) ∪ (0, ∞)
(b) Hole(s):
None
(c) x-intercepts: x = −3, x = −2, x = 3
(d) y-intercept: None
(e) Vertical Asymptote:
x=0
(f) Horizontal Asymptote(s):
(g) Slant Asymptote:
None
y = x+2
47. (a) y = 4
19
(b) x =
16
19
(c) , 4
16
49. (a)
(1, 0 )
6
4
2
x
−8
Continued in the next column…
y
(b)
−6
−4
−2
2
4
6
8
−2
−4
−6
150
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 2.4: Applications and Writing Functions
1.
(a) A ( w ) = 27 w − w2
27
= 13.5 ft.
2
(c) The maximum area of the rectangle is
729
A=
= 182.25 ft 2 .
4
(b) A is greatest when w =
3.
100
7.
16
9.
33. A ( r ) = 4r 2
(b) A ( w ) = w 64 − w2
37. (a) A ( x ) =
375 ft × 750 ft (where the 750 ft side is parallel to
the river)
11. 20,000 ft 2
13. A ( x ) =
(b) d ( x ) = x 4 − 15 x 2 + 64
35. (a) P ( w ) = 2w + 2 64 − w2
d ( t ) = 130t
5.
31. (a) d ( x ) = x 4 − 19 x 2 + 100
4 Px − x 2
4
P
P
(b) Domain of A:
<x<
4
2
39. (a) d ( t ) = 2500t 2 − 360t + 13
x
36 − x 2
2
15. P ( ) = 2 +
( P − 2x )
44 2 2 + 44
=
17. (a) A ( x ) = 18 x − 2 x3
(b) Note: To minimize d ( t ) , minimize the
quadratic under the radical sign.
9
t=
hr = 0.072 hr = 4.32 min
125
41. d = 20 2 cm ≈ 28.28 cm
Note: See the hint in the solutions for numbers 39
and 40 for minimizing the function.
(b) P ( x ) = 4 x + 18 − 2 x 2
43. C ( x ) =
19. V ( r ) = 2π r 3
2π x 2π x 3
=
3
3
600
r
600
2π r 3 + 600
(b) S ( r ) =
+ 2π r 2 =
r
r
21. (a) L ( r ) =
23. V ( r ) =
4π r 3
3
25. S ( x ) = x 2 +
80 x3 + 80
=
x
x
27. (a) S ( x ) = 120 x − 6 x 2
(b) 10 cm × 10 cm × 10 cm
(c) 1000 cm2
29. (a) C ( x ) = x
(b) A ( x ) =
x2
4π
MATH 1330 Precalculus
151
Odd-Numbered Answers to Exercise Set 3.1: Exponential Equations and Functions
1.
(a) No solution
3.
(a) x =
5.
(a) x = 1
7.
(a) x = −
4
3
(b) x =
7
2
(b) x =
5
6
13. (a)
(b) No solution
29
6
(b) x = −
1
4
9.
Horizontal Asymptote at y = 0
Passes through ( 0, 1)
(b)
x
0
f ( x)
1
1
3
−1
1
3
Horizontal Asymptote at y = 0
Horizontal
Asymptote
at y = 0
Passes through ( 0, 1)
(c) The two graphs have the point ( 0, 1) in
common.
(d) Both graphs have a horizontal asymptote at
y = 0.
11.
15. (a)
Horizontal Asymptote at y = 0
x
0
f ( x)
1
1
1
2
−1
2
Horizontal
Asymptote
at y = 0
Both graphs pass through ( 0, 1)
(b) To obtain the graph of g , reflect the graph of f
across the y-axis.
Continued on the next page…
152
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 3.1: Exponential Equations and Functions
x
x
1
(c) g ( x ) = = 6 −1 = 6− x . Therefore,
6
g ( x ) = f ( − x ) . It is known from previous
( )
lessons on transformations that to obtain the
graph of g ( x ) = 6− x , the graph of f ( x ) = 6 x is
23. (a) x-intercept(s):
y-intercept:
None
1
2
(b)
reflected in the y-axis.
17.
(c) Domain:
Range:
(− ∞, ∞ )
(0, ∞ )
(d) Increasing
19.
25. (a) x-intercept(s):
y-intercept:
None
−1
(b)
21.
(c) Domain:
Range:
(− ∞, ∞ )
(− ∞, 0)
(d) Decreasing
MATH 1330 Precalculus
153
Odd-Numbered Answers to Exercise Set 3.1: Exponential Equations and Functions
27. (a) x-intercept(s):
y-intercept:
None
1
1
2
y-intercept: 54
31. (a) x-intercept:
(b)
−
(b)
(c) Domain:
Range:
(− ∞, ∞ )
(0, ∞ )
Note: The graph passes through ( −2, − 26 ) .
(c) Domain:
(d) Decreasing
Range:
29. (a) x-intercept:
y-intercept:
−2
−3
(− ∞, ∞ )
( −27, ∞ )
(d) Increasing
(b)
33. (a) x-intercept:
y-intercept:
−
1
2
−2
(b)
(c) Domain:
Range:
(− ∞, ∞ )
( −4, ∞ )
(d) Decreasing
(c) Domain:
Range:
(− ∞, ∞ )
( −∞, 2 )
Continued in the next column…
(d) Decreasing
35. (a) x-intercept:
154
−2
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 3.1: Exponential Equations and Functions
y-intercept:
−72
39. (a) x-intercept:
(b)
y-intercept:
8
− , 0
3
( 0, − 31 78 )
(b)
Note: The graph passes through ( −4, 8 ) .
(c) Domain:
Range:
(− ∞, ∞ )
Note: The graph passes through ( −1, − 31) .
( −∞, 9 )
(d) Decreasing
(c) Domain:
Range:
( −∞, ∞ )
( −32, ∞ )
(d) Decreasing
37. (a) x-intercept: None
y-intercept: 9
41. (a) x-intercept(s):
y-intercept:
None
−37
(b)
(b)
(c) Domain: (− ∞, ∞ )
Range: (0, ∞ )
(d) Decreasing
The graph passes through the point ( 0, − 37 ) .
(c) Domain:
Range:
( −∞, ∞ )
( −∞, − 36 )
(d) Increasing
MATH 1330 Precalculus
155
Odd-Numbered Answers to Exercise Set 3.1: Exponential Equations and Functions
43. (a) x-intercept: None
y-intercept: 2
61. Graph:
(b)
(a)
(b)
(c)
(d)
(c) Domain: (− ∞, ∞ )
Range: (0, ∞ )
(d) Increasing
Domain: (− ∞, ∞ )
Range: (0, ∞ )
Horizontal Asymptote at y = 0
y-intercept: 1
63.
45. f ( x) = 4 ⋅ 2 x
47. f ( x) = 2 ⋅ 7 x
49. True
51. True
53. False
55. False
Domain: (− ∞, ∞ ) ;
Range: (− ∞, 0)
Domain: (− ∞, ∞ ) ;
Range: (− 6, ∞ )
57. True
59.
65.
156
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 3.1: Exponential Equations and Functions
67.
Domain: (− ∞, ∞ ) ;
Range: ( 3, ∞ )
Domain: (− ∞, ∞ ) ;
Range: (− ∞, − 5)
Domain: (− ∞, ∞ ) ;
Range: (− 3, ∞ )
69.
71.
MATH 1330 Precalculus
157
Odd-Numbered Answers to Exercise Set 3.2: Logarithmic Functions
1.
(a) 34 = 81
(b) 26 = 64
3.
(a) 5x = 7
(b) e8 = x
5.
(a) log 6 (1) = 0
(b) log10
(
1
10,000
2
45. (a) x = log ≈ −0.398
5
2
(b) x = ln = −0.916
5
) = −4
7.
(a) log 7 ( 7 ) = 1
(b) ln ( x ) = 4
9.
(a) ln ( 7 ) = x
(b) ln ( y ) = x − 2
11. (a) 3
(b) 0
13. (a) 1
(b) 2
47. x =
2 + log ( 6 )
≈ 0.926
3
49. x = ±
ln ( 40 )
≈ ± 0.960
2
51. (a) and (b): See graph below
f
y
8
1
15. (a)
2
(b) −1
1
3
(b) −2
17. (a)
6
f -1
4
2
x
−4
−2
2
4
6
8
−2
1
4
19. (a) −1
(b)
1
21. (a) −
2
5
(b)
3
23. (a) 1
(b) 4
25. (a) −6
(b)
−4
(c) The inverse of y = 2 x is y = log 2 ( x ) .
53. (a) Graph:
y
1
3
x
−2
27. (a) −4
(b) x
29. (a) 5
(b)
3
(b) x
4
0
2
4
6
8
Note: The scale on
the y-axis varies,
depending on the
value of b.
(b) The x-intercept is 1. There are no y-intercepts.
31. (a) 6
(c) There is a vertical asymptote at x = 0 .
33. (a) x = 5
(b) x = 3
35. (a) x = 49
(b) x = 9
55. (a)
y
6
4
3
37. (a) x =
2
(b) x = 2
2
x
39. (a) x = −97; x = 103
(b) x = 4
−2
2
4
6
8
−2
41. (a) x = e
4
(b) x = e − 5
43. (a) x = log ( 20 ) ≈ 1.301
(b) x = ln ( 20 ) ≈ 2.996
158
Vertical asymptote at x = 0 .
Graph passes through (1, 3)
2
(b) Domain:
( 0, ∞ ) ;
Range:
( −∞, ∞ )
(c) Increasing
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 3.2: Logarithmic Functions
57. (a)
63. (a)
6 y
y
2
4
x
−4
2
−2
2
x
−2
2
4
4
Vertical Asymptote at x = 0
Graph passes through ( −1, 0 )
−2
6
−4
−2
Vertical Asymptote at x = 0
Graph passes through (1, 0 )
(b) Domain: ( −∞, 0 )
Range:
(b) Domain: ( 0, ∞ )
( −∞, ∞ )
Range:
( −∞, ∞ )
(c) Decreasing
(c) Decreasing
y
65. (a)
4
59. (a)
Vertical Asymptote at x = −4
Graph passes through ( −3, 0 )
y
2
2
x
−6
−4
−2
2
4
x
−2
−2
2
4
6
8
−4
−2
Vertical Asymptote at x = 5
Graph passes through ( 6, 0 )
(b) Domain: ( 5, ∞ )
Range:
(b) Domain: ( −4, ∞ )
Range:
( −∞, ∞ )
(c) Decreasing
( −∞, ∞ )
(c) Increasing
y
67. (a)
6
61. (a)
Vertical Asymptote at x = 0
Passes through ( −1, 5)
4
y
2
2
x
−2
2
x
4
−4
−2
−2
2
4
6
−2
−4
−6
Vertical Asymptote at x = −1
Graph passes through ( 0, − 4 )
(b) Domain: ( −∞, 0 )
Range:
(b) Domain: ( −1, ∞ )
Range:
( −∞, ∞ )
( −∞, ∞ )
(c) Increasing
(c) Increasing
MATH 1330 Precalculus
159
Odd-Numbered Answers to Exercise Set 3.2: Logarithmic Functions
69.
( 0, ∞ )
71.
( 0, ∞ )
73.
( −2, ∞ )
75.
( −∞, 0 )
5
77. −∞,
2
79.
( −∞, ∞ )
81.
( −∞, − 2 )
160
∪ ( 3, ∞ )
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 3.3: Laws of Logarithms
1.
True
3.
False
5.
False
7.
False
9.
True
K 4 P3
41. log
Q
3
x − 2)
(
43. log 7
5 x 2 − 3 ( x + 5 )4
45. log 7 ( 40 )
11. False
(
47. ln x 2 + 5 x + 6
13. False
15. True
)
2
6 5
x ( x + 3)
49. ln
4
x2 + 5
17. True
(
)
19. False
51. 3
21. log5 ( 9 ) + log 5 ( C ) + log 5 ( D )
23. log 7 ( K ) + log 7 ( P ) − log 7 ( L )
53. 2 14 , or
9
4
55. 3
25. 2 ln ( B ) + 3ln ( P ) + 16 ln ( K )
57. 3
27. log ( 9 ) + 2log ( k ) + 3log ( m ) − 4log ( p ) − log ( w )
31.
1
log3
4
( 7 ) + 14 log 3 ( x )
33. 4 ln ( x ) + ln( x − 5) − 12 ln( x + 3)
35.
59.
3
2
61.
3
2
( x ) + ln( y + 3)
29.
1
ln
2
1
log
2
(x
2
)
(
)
− 7 − 12 log x 2 + 3 − log ( x − 4 )
AB
37. (a) log 5
C
(b) 1 = log 5 ( 5 )
5 AB
(c) log5
C
AB
(d) log 5
5C
YW
39. ln
Z
MATH 1330 Precalculus
63. 25
65. 81
67. 34
69. 320
71. 30
73. y = x3
75. y =
7
x5
x4 ( x + 2)
77. y =
7
3
161
Odd-Numbered Answers to Exercise Set 3.3: Laws of Logarithms
79. (a)
(b)
81. (a)
(b)
ln ( 2 )
≈ 0.431
ln ( 5 )
ln (17.2 )
≈ 1.588
ln ( 6 )
log ( 4 )
≈ 0.631
log ( 9 )
log ( 8.9 )
≈ 1.577
log ( 4 )
83. 3
85. 24
87. (a) ln ( 2 ) + ln ( 3)
(b) 2 log ( 2 ) + log ( 7 )
89. (a) 2 log ( 2 ) + log ( 3) + log ( 5 )
(b) 3ln ( 3) + ln ( 2 ) + ln ( 5 )
91. (a) A + C
(b) − B
93. (a) A − C
(b) 2C
95. (a) 2B + C
(b) 3B − 2 A − C
97. (a)
2
1
A+ B
3
3
(b) 2 B − 2C − 2 A
99. (a) 2 A + 2C + 1
(b)
162
C
B
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 3.4:
Exponential and Logarithmic Equations and Inequalities
1.
(a) x =
ln (12 )
ln ( 7 )
(b) x =
2ln ( 2 ) + ln ( 3)
ln ( 7 )
(c) x ≈ 1.277
3.
(a) x = ln ( 22 )
15. (a) x =
(b) x =
5.
(a) x =
ln (14 )
ln ( 6 )
ln ( 2 ) + ln ( 7 )
(b) x =
ln ( 2 ) + ln ( 3)
(c) x ≈ 1.473
7.
(a) – (c) No solution
9.
(a) x = −
ln ( 7 )
3ln (19 )
(b) x = −
ln ( 7 )
3ln (19 )
17. (a) x =
(b) x =
19. (a) x =
(c) x ≈ 0.773
13. (a) x =
(b) x =
ln ( 36 ) − 4
5
2 ln ( 2 ) + 2ln ( 3) − 4
5
(c) x ≈ −0.083
6ln ( 2 ) + 3ln ( 5 )
5ln ( 2 )
− ln ( 6 )
ln ( 6 )
=
4ln ( 5 ) − 7 ln ( 6 ) 7 ln ( 6 ) − 4ln ( 5 )
(b) x =
− ln ( 2 ) − ln ( 3)
4ln ( 5 ) − 7 ln ( 2 ) − 7 ln ( 3)
=
ln ( 2 ) + ln ( 3)
7 ln ( 2 ) + 7 ln ( 3) − 4 ln ( 5 )
(c) x ≈ 0.294
21. (a) x =
(b) x =
(b) x = ln (13) − ln ( 2 ) − ln ( 3)
3ln ( 20 )
ln ( 32 )
(c) x ≈ 2.593
(c) x ≈ −0.220
13
11. (a) x = ln
6
ln ( 2 ) + ln ( 5 ) − 4ln ( 7 )
ln ( 7 )
(c) x ≈ −2.817
(b) x = ln ( 2 ) + ln (11)
(c) x ≈ 3.091
ln (10 ) − 4 ln ( 7 )
ln ( 7 )
ln ( 2 ) + 5ln ( 8 )
3ln ( 2 ) − 2ln ( 8 )
16ln ( 2 )
16 16
=−
is the final answer
−
−3ln ( 2 )
3 3
(c) x ≈ −5.333
−1
16
16
7
23. (a) x = − ln = ln = ln
7
7
16
(b) x = ln ( 7 ) − 4 ln ( 2 )
(c) x ≈ −0.827
25. (a) x = ln ( 8)
(b) x = 3ln ( 2 )
(c) x ≈ 2.079
27. (a) – (c) x = 1
MATH 1330 Precalculus
163
Odd-Numbered Answers to Exercise Set 3.4:
Exponential and Logarithmic Equations and Inequalities
29. (a) x =
(b) x =
ln ( 3 )
ln ( 7 )
ln ( 3 )
ln ( 7 )
(c) x ≈ 0.565
57. x = 3
36
is not a solution; if it is
7
plugged into the original equation, the resulting
equation contains logarithms of negative numbers,
which are undefined.)
59. No solution. (Note: x = −
27
2
31. (a) – (c) x = −3, x = 3
61. x =
33. (a) – (c) x = −1, x = 5
63. x = 6 (Note: x = −4 is not a solution; if it is plugged
into the original equation, the resulting equation
contains the logarithms of negative numbers, which
are undefined.)
7
3
Note: The answer is not all real numbers. Recall the
domain of log 2 ( 3 x − 7 ) . The inner portion of a
35. (a) – (c) x >
logarithmic function must be positive, i.e. 3x − 7 > 0 .
=e
1
5
e
67. x = 5
37. x = e8
39. x = e −7 =
65. x = e
e −5
1
e7
69. x > 0 (Note: The answer is not all real numbers.
Recall the domain of ln ( x ) . The inner portion of a
logarithmic function must be positive, i.e. x > 0 .)
41. x = 1,000
43. x = 24
45. x =
47. x =
101
4
e9 − 8
3
49. x = 6 (Note: x = −1 is not a solution; if it is
plugged into the original equation, the resulting
equation contains logarithms of negative numbers,
which are undefined.)
71. No solution. (Note: x = 0 is not a solution; if it is
plugged into the original equation, the resulting
equation contains the logarithm of zero, which is
undefined.)
73. x = 5 (Note: x = −5 is not a solution; if it is
plugged into the original equation, the resulting
equation contains the logarithm of a negative
number, which is undefined.)
75. (a) The following answers are equivalent; either one
is acceptable:
x>
51. x = 4 (Note: x = −2 is not a solution; if it is
plugged into the original equation, the resulting
equation contains logarithms of negative numbers,
which are undefined.)
53. x = −1 (Note: x = −4 is not a solution; if it is
plugged into the original equation, the resulting
equation contains logarithms of negative numbers,
which are undefined.)
55. x =
164
28
3
ln (15 ) − 3ln ( 8 )
ln ( 8)
; x>
ln (15)
ln ( 8 )
−3
(b) The following answers are equivalent; either one
is acceptable:
x>
x>
ln ( 5 ) + ln ( 3) − 9 ln ( 2 )
3ln ( 2 )
ln ( 5 ) + ln ( 3)
3ln ( 2 )
−3
Continued on the next page…
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 3.4:
Exponential and Logarithmic Equations and Inequalities
Interval Notation:
ln ( 5 ) + ln ( 3) − 9 ln ( 2 )
, ∞
3ln ( 2 )
ln ( 5) + ln ( 3)
Also acceptable:
− 3, ∞
3ln ( 2 )
125
ln
8
83. (a) x <
ln ( 0.4 )
Note: ln ( 0.4 ) is negative; dividing by ln ( 0.4 )
has reversed the inequality symbol.
(b) Solution: x < −3
(c) x > −1.698 ; Interval Notation: ( −1.698, ∞ )
77. (a) The solution is all real numbers. We cannot take
the logarithm of both sides of the inequality
7 x > 0 , since ln ( 0 ) is undefined. Regardless of
the value of x, the function f ( x ) = 7 x is always
positive. Therefore, the solution is all real
numbers.
(b) All real numbers; ( −∞, ∞ )
(c) All real numbers; ( −∞, ∞ )
The final simplification steps for this problem
are shown below:
x<
3ln ( 5 ) − 3ln ( 2 )
ln ( 2 ) − ln ( 5 )
3 ln ( 5 ) − ln ( 2 )
x<
ln ( 2 ) − ln ( 5 )
Note :
ln ( 5 ) − ln ( 2 )
ln ( 2 ) − ln ( 5 )
= −1
Therefore, x < −3.
Interval Notation:
( −∞, − 3)
(c) x < −3; Interval Notation:
( −∞, − 3)
79. (a) x ≥ 2 − ln ( 9 )
(b) x ≥ 2 − 2 ln ( 3)
Interval Notation: 2 − 2 ln ( 3) , ∞ )
(c) x ≥ −0.197 ; Interval Notation: [ −0.197, ∞ )
81. (a) x >
ln ( 7 )
ln ( 0.2 )
Note: ln ( 0.2 ) is negative; dividing by ln ( 0.2 )
has reversed the inequality symbol.
(b) x > −
ln ( 7 )
ln ( 5 )
ln ( 7 )
Interval Notation: −
, ∞
ln ( 5 )
(c) x > −1.209 ; Interval Notation: ( −1.209, ∞ )
85. (a) No solution
Note: The solution process yields the equation
46
ex ≤ −
. There is no solution, since e x
5
is always positive.
(b) No solution
(c) No solution
87. (a) Solution: x ≥ 1
Explanation:
First, find the domain of log 2 ( x ) . The domain
of log 2 ( x ) is x > 0 .
Next, isolate x in the given inequality:
log 2 ( x ) ≥ 0
2
log 2 ( x )
≥ 20
x ≥1
The solution must satisfy both x > 0 and x ≥ 1 .
The intersection of these two sets is x ≥ 1 , which
is the solution to the inequality.
Continued on the next page…
MATH 1330 Precalculus
165
Odd-Numbered Answers to Exercise Set 3.4:
Exponential and Logarithmic Equations and Inequalities
(b) x ≥ 1 ; Interval Notation: [1, ∞ )
(c) x ≥ 1 ; Interval Notation: [1, ∞ )
89. (a) Solution: 2 ≤ x <
7
3
Explanation:
First find the domain of ln ( 7 − 3x ) :
7 − 3x > 0
−3x > −7
7
x<
3
Next, isolate x in the given inequality:
ln ( 7 − 3 x ) ≤ 0
)
e (
≤ e0
7 − 3x ≤ 1
−3 x ≤ −6
x≥2
ln 7 − 3 x
7
and x ≥ 2 .
3
7
The intersection of these two sets is 2 ≤ x < ,
3
which is the solution to the inequality.
The solution must satisfy both x <
166
(b) 2 ≤ x <
7
; Interval Notation:
3
7
2, 3
(c) 2 ≤ x <
7
; Interval Notation:
3
2,
7
3
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 3.5: Applications of Exponential Functions
1.
A(t) = the total amount of money in the investment
after t years
P = the principal, i.e. the original amount of money
invested
r = the interest rate, expressed as a decimal (rather
than a percent)
n = the number of times that the interest is
compounded each year
t = the number of years for which the money is
invested
3.
(a) P
(b) A(t )
5.
(a)
(b)
(c)
(d)
(e)
7.
$10,133.46
9.
6.14%
27. (a) m(t ) = 75e −0.0139t
(b) 24.67 grams
(c) 144.96 years
$5,901.45
$5,969.37
$6,004.79
$6,028.98
$6,041.26
11. Option (a) is a better investment (7.5%, compounded
quarterly).
13. (a)
(b)
(c)
(d)
(e)
250 rabbits
2% per day
271 rabbits
380 rabbits
34.66 days
15. (a) N (t ) = 18,000e0.75t
(b) 7,261,718 bacteria
(c) 2.27 hours
17. 3,113 people
19. (a) 317 buffalo
(b) 387 buffalo
(c) 57.44 years
21. (a)
(b)
(c)
(d)
70 grams
27.62 grams
39.46%
11.18 days
23. (a) $28,319.85
(b) 13.95 years
25. (a) $17,926.76
(b) 5.60 years
MATH 1330 Precalculus
167
Odd-Numbered Answers to Exercise Set 4.1:
Special Right Triangles and Trigonometric Ratios
1.
angles
3.
180
(
15. (a) x = 8 2
(b)
5.
largest, smallest
7.
(a) x = 5 2
)
2, so x = 16
2
(8 2 ) + (8 2 )
2
= x2
64 ( 2 ) + 64 ( 2 ) = x 2
128 + 128 = x 2
(b) 52 + 52 = x 2
256 = x 2
25 + 25 = x 2
x = 16
50 = x 2
x = 50 = 25 2
17. (a) x =
2 3 2 3 2 2 6
=
⋅
=
, so x = 6
2
2
2
2
x=5 2
(
(b) x 2 + x 2 = 2 3
9.
(a) x =
4 2
, so x = 4
2
(
(b) x 2 + x 2 = 4 2
)
x 2 = 16
x=4
11. (a) x =
8
8
2 8 2
=
⋅
=
, so x = 4 2
2
2
2 2
2
2 x 2 = 4 ( 3) = 12
x2 = 6
2
2 x 2 = 16 ( 2 ) = 32
)
x= 6
19. 60
21. (a) a = b = 5
(b) The altitude divides the equilateral triangle into
two congruent 30o-60o-90o triangles. Therefore,
a = b . Since a + b = 10 , then a + a = 10 , so
a = 5 . Hence a = b = 5 .
(b) x 2 + x 2 = 82
2 x 2 = 64
(c) 52 + c 2 = 102
x 2 = 32
25 + c 2 = 100
x = 32 = 16 2
x=4 2
c 2 = 75
c = 75 = 25 3
c=5 3
13. (a) x =
9
9
2
9 2
=
⋅
, so x =
2
2
2 2
(b) x 2 + x 2 = 92
2 x 2 = 81
81
x2 =
2
81
9
9
2
x=
=
=
⋅
2
2
2 2
x=
168
9 2
2
23. x = 14, y = 7 3
25. x = 12, y = 6
5
5 3
27. x = , y =
2
2
29. x = 4 3, y = 2 3
31. x = 15, y = 10 3
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 4.1:
Special Right Triangles and Trigonometric Ratios
33. (a)
( BC )2 + 152 = 17 2
2
( BC ) + 225 = 289
( BC )2 = 64
45. (a) 42 + 82 = x 2
16 + 64 = x 2
x 2 = 80
x = 80 = 16 5
BC = 8
x=4 5
8
(b) sin ( A) =
17
cos ( A ) =
15
17
8
tan ( A ) =
15
35. cos (θ ) =
15
sin ( B ) =
17
cos ( B ) =
8
17
15
tan ( B ) =
8
2 6
5 6
, tan (θ ) =
7
12
(b) sin (α ) =
2 5
5
5
5
cos (α ) =
5
2
csc (α ) =
sec (α ) = 5
tan (α ) = 2
cot (α ) =
1
2
5
5
csc ( β ) = 5
cos ( β ) =
2 5
5
sec ( β ) =
tan ( β ) =
1
2
cot ( β ) = 2
(c) sin ( β ) =
37. cosecant
39. cotangent
41. cosine
5
2
43. (a) x 2 + 52 = 62
x 2 + 25 = 36
x = 11
(b) sin (α ) =
5
6
csc (α ) =
6
5
cos (α ) =
11
6
sec (α ) =
6 11
11
tan (α ) =
5 11
11
cot (α ) =
11
5
11
6
csc ( β ) =
6 11
11
sec ( β ) =
6
5
cot ( β ) =
5 11
11
(c) sin ( β ) =
cos ( β ) =
tan ( β ) =
5
6
11
5
3
7
csc (θ ) =
7
3
cos (θ ) =
2 10
7
sec (θ ) =
7 10
20
tan (θ ) =
3 10
20
cot (θ ) =
2 10
3
47. sin (θ ) =
x 2 = 11
MATH 1330 Precalculus
Continued on the next page…
169
Odd-Numbered Answers to Exercise Set 4.1:
Special Right Triangles and Trigonometric Ratios
49. (a)
3 2
4
3
45
o
30
60o
2 3
2
2
csc 45 = 2
( ) 22
tan ( 45 ) = 1
sec 45 = 2
( )
cos 45 =
( )
(c) sin 30 =
( )
cos 30 =
1
2
2
o
3
(b) sin 45 =
( )
( )
cot ( 45 ) = 1
( )
3
tan 30 =
3
( )
3
(d) sin 60 =
2
( )
1
cos 60 =
2
( )
sec 30 =
2 3
3
( )
cot 30 = 3
2 3
csc 60 =
3
( )
sec 60 = 2
( )
( )
cot 60 =
tan 60 = 3
( )
In addition to the observations above, the following
formulas, known as the cofunction identities, are
illustrated in parts (b) – (d), and will be covered in
more detail in Chapter 6.
(
)
e.g. sin ( 30 ) = cos ( 60 )
sin ( 60 ) = cos ( 30 )
sin ( 45 ) = cos ( 45 )
(
)
e.g. sec ( 30 ) = csc ( 60 )
sec ( 60 ) = csc ( 30 )
sec ( 45 ) = csc ( 45 )
sec (θ ) = csc 90 − θ ,
(
)
e.g. tan ( 30 ) = cot ( 60 )
tan ( 60 ) = cot ( 30 )
tan ( 45 ) = cot ( 45 )
tan (θ ) = cot 90 − θ ,
( )
( )
sin (θ ) = cos 90 − θ ,
csc 30 = 2
3
2
1
in both exercises. In fact, the
2
1
value of cos 60 is always , and the
2
trigonometric ratio of any given number is
constant.
( )
that cos 60 =
3
3
51. Answers vary, but some important observations are
below.
First, notice that when comparing Exercises 49 and
50:
The answers to part (b) are identical, for each
respective trigonometric function. So, for
2
in both examples.
2
Similarly, the answers to part (c) are identical.
( )
example, sin 45 =
3
regardless of
3
the triangle being used to compute the ratio.
Along the same lines, the answers to part (d) in
both exercises are identical. Notice, for example,
( )
So, for example, tan 30 =
170
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 4.2:
Radians, Arc Length, and the Area of a Sector
1.
θ =4
3.
θ = 240
25. s = 5π ft
21π
yd
2
27. s =
5.
7.
θ=
1
2
29. s = 2π ft
(a) There are slightly more than 6 radians in one
revolution, as shown in the figure below.
2 radians
1 radian
3 radians
0 radians
6 radians
4 radians
Justification:
The arc length in this case is the circumference C
of the circle, so s = C = 2π r . Therefore,
s 2π r
θ= =
= 2π .
r
r
Comparison to part (a):
2π ≈ 6.28 , and it can be seen from part (a) that
there are slightly more than 6 radians in one
revolution.
(a)
π
6
≈ 0.52
(b)
; s = 4π cm (Notice that s = 4π cm in the
3
solution for Exercise 23.)
5π
; s = 5π ft (Notice that s = 5π ft in the
4
solution for Exercise 25.)
33. θ =
5 radians
(b) There are 2π radians in a complete revolution.
9.
π
2
≈ 1.57
(c)
35. s =
13. (a)
2π
≈ 2.09
3
5π
≈ 3.93
4
(c)
19π
2π
≈ 0.33 (b)
≈ 0.70
180
9
(c)
(b)
39. r =
24
m = 4.8 m
5
41. r =
28
ft ≈ 2.97 ft
3π
43. r =
12
in = 2.4 in
5
3π
≈ 2.36
4
11π
≈ 5.76
6
2π
≈ 1.26
5
15. (a) 45
(b) 240
(c) 150
17. (a) 180
(b) 165
(c) 305
15π
in
2
37. s = 10π cm
45. r =
11. (a)
π
31. θ =
10
m ≈ 3.18 m
4π
47. P =
+ 16 cm ≈ 20.19 cm
3
49. A = 24π cm 2
51. A = 10π ft 2
53. A =
19. (a) 143.24
π
189π
yd 2
4
(b) 28.99
55. A = 2π ft 2
21.
31π
60
23. s = 4π cm
MATH 1330 Precalculus
57. θ = 3; A = 24π cm 2 (Notice that A = 24π cm 2 in
the solution for Exercise 49.)
171
Odd-Numbered Answers to Exercise Set 4.2:
Radians, Arc Length, and the Area of a Sector
5π
; A = 10π ft 2 (Notice that A = 10π ft 2 in the
4
solution for Exercise 51.)
59. θ =
61. r = 7 cm
63. r =
65. θ =
4
in
5
(b) Three methods of solution are shown below:
Method 1: (shown in the text)
The linear speed v can be found using the
d
formula v = , where d is the distance traveled
t
by the point on the CD in time t.
Since the rate of turn is 900 / sec =
7π
4
900
, we
1 sec
note that θ = 900 and t = 1 sec .
67. (a) Two methods of solution are shown below.
Method 1: (shown in the text)
The angular speed ω can be found using the
formula ω =
θ
, where θ is in radians. Since the
t
rate of turn is 900 / sec =
900
, we note that
1 sec
To find d, use the formula d = rθ (equivalent to
the formula for arc length, s = rθ ). We must
first convert θ = 900 to radians:
θ = 900 ×
5
π
180
= 900 ×
π
180
= 5π
Then d = rθ = ( 6cm )( 5π ) = 30π cm .
θ = 900 and t = 1 sec .
Finally, use the formula for linear speed:
d 30π cm
= 30π cm/sec
v= =
1 sec
t
First, convert 900 to radians:
θ = 900 ×
Then ω =
5
π
180
θ
t
=
= 900 ×
π
180
= 5π
5π radians
= 5π radians/sec
1 sec
Method 2:
Take the given rate of turn, 900 / sec , equivalent
900
to
, and use unit conversion ratios to
1 sec
change this rate of turn from degrees/sec to
radians/sec, as shown below:
Method 2:
Another formula for the linear speed v is v = rω ,
where ω is the angular speed. (This can be
derived from previous formulas, since
d rθ
θ
v= =
= r = rω .)
t
t
t
It is given that r = 6 cm , and it is known from
part (a) that w = 5π radians/sec, so:
v = rw = ( 6 cm )( 5π radians/sec ) = 30π cm/sec .
(Remember that units are not written for radians,
with the exception of angular speed.)
5
900 π radians 900 π radians
×
=
×
1 sec
1 sec
180
180
= 5π radians/sec
Method 3:
Take the given rate of turn, 900 / sec , equivalent
900
to
, and use unit conversion ratios to
1 sec
change this rate of turn from degrees/sec to
cm/sec, as shown below.
Continued on the next page…
172
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 4.2:
Radians, Arc Length, and the Area of a Sector
(Note that in one revolution, the circumference
of the circle with a 6 cm radius is
C = 2π r = 2π ( 6 ) = 12π cm , hence the unit
conversion ratio
12π cm
.
1 revolution
900 π radians 1 revolution
12π cm
×
×
×
1 sec
2π radians 1 revolution
180
5
6
12π cm
900 π radians 1 revolution
=
×
×
×
1 sec
2π radians 1 revolution
180
= 30π cm/sec
(c) This problem can be solved in one of three ways:
Method 1:
The linear speed v can be found using the
d
formula v = , where d is the distance traveled
t
by the point on the CD in time t.
Since the rate of turn is 900 / sec =
900
, we
1 sec
derived from previous formulas, since
d rθ
θ
v= =
= r = rω .)
t
t
t
The CD has a radius of 6 cm, so the desired
radius of the point halfway between the center of
1
the CD and its outer edge is r = ( 6 ) = 3 cm .
2
It is known from part (a) that w = 5π
radians/sec, so:
v = rw = ( 3 cm )( 5π radians/sec ) = 15π cm/sec .
(Remember that units are not written for radians,
with the exception of angular speed.)
Method 3:
Take the given rate of turn, 900 / sec , equivalent
900
to
, and use unit conversion ratios to
1 sec
change this rate of turn from degrees/sec to
cm/sec, as shown below.
(Note that in one revolution, the circumference
of the circle with a 3 cm radius is
C = 2π r = 2π ( 3) = 6π cm , hence the unit
conversion ratio
note that θ = 900 and t = 1 sec .
To find d, use the formula d = rθ (equivalent to
the formula for arc length, s = rθ ). We must
first convert θ = 900 to radians:
θ = 900 ×
π
180
5
= 900 ×
π
180
= 5π
6π cm
.
1 revolution
900 π radians 1 revolution
6π cm
×
×
×
1 sec
2π radians 1 revolution
180
5
3
900 π radians 1 revolution
6π cm
=
×
×
×
1 sec
2π radians 1 revolution
180
= 15π cm/sec
The CD has a radius of 6 cm, so the desired
radius of the point halfway between the center of
1
the CD and its outer edge is r = ( 6 ) = 3 cm .
2
Then d = rθ = ( 3cm )( 5π ) = 15π cm .
Finally, use the formula for linear speed:
d 15π cm
v= =
= 15π cm/sec
t
1 sec
69. Since the diameter is 26 inches, the radius, r = 13 in.
Take the given rate of turn, 20 miles/hr, equivalent to
20 miles
, and use unit conversion ratios to change
1 hr
this rate of turn from miles/hr to rev/min, as shown
below.
Continued on the next page…
Method 2:
Another formula for the linear speed v is v = rω ,
where ω is the angular speed. (This can be
MATH 1330 Precalculus
173
Odd-Numbered Answers to Exercise Set 4.2:
Radians, Arc Length, and the Area of a Sector
73. Hour Hand:
r = 4 in
(Note that in one revolution, the circumference of the
circle with a 13 in radius is
C = 2π r = 2π (13) = 26π in , hence the unit
conversion ratio
Rate of turn:
1 revolution
.
26π in
(Note that in one revolution, the circumference of the
circle with a 4 in radius is C = 2π r = 2π ( 4 ) = 8π in ,
20 miles 5280 ft 12 in
1 hr
1 revolution
×
×
×
×
1 hr
1 mile 1 ft 60 min
26π in
=
hence the unit conversion ratio
1 hr
1 revolution
20 miles 5280 ft 12 in
×
×
×
×
60
min
1 hr
1 mile
1 ft
26π in
20 min ×
≈ 258.57 rev/min, or 258.57 rpm
=
(a) Take the given rate of turn, 4 revolutions per
4 revolutions
, and
second, equivalent to
1 sec
use unit conversion ratios to change this rate
of turn from revolutions/sec to radians/sec,
as shown below.
1 revolution
.
8π in
1 hr
1 revolution
8π in
×
×
60 min
12 hr
1 revolution
= 20 min ×
71. Numbers 67 and 68 show various methods for solving
these types of problems. One method is shown below
for each question.
1 hr
1 revolution
8π in
×
×
60 min
12 hr
1 revolution
2π
in
9
Minute Hand:
r = 5 in
Rate of turn:
1 revolution
60 min
(Note that in one revolution, the circumference of the
circle with a 5 in radius is
C = 2π r = 2π ( 5 ) = 10π in , hence the unit
4 revolutions 2π radians
×
1 second
1 revolution
=
1 revolution
12 hours
4 revolutions
2π radians
×
1 second
1 revolution
conversion ratio
1 revolution
.
10π in
= 8π radians/sec
(b) v = rω = (10 in )( 8π radians/sec ) , so
20 min ×
v = 80π in/sec
= 20 min ×
(c) Use the linear speed from part (b),
v = 80π in/sec , and then use unit conversion
ratios to change this from in/sec to miles/hr,
as shown below:
=
80π in 1 ft
1 mile 60 sec 1 min
1
×
×
×
×
×
1 sec 12 in 5280 ft
1
60 sec 60 min
80π in 1 ft
1 mile 60 sec 60 min
×
×
×
×
1 hr
1 sec 12 in 5280 ft 1 min
= 14.28 miles per hour (mph)
=
1 revolution
10π in
×
60 min
1 revolution
1 revolution
10π in
×
60 min
1 revolution
10π
in
3
Second Hand:
r = 6 in
Rate of turn:
1 revolution
1 min
Continued on the next page…
174
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 4.2:
Radians, Arc Length, and the Area of a Sector
(Note that in one revolution, the circumference of the
circle with a 6 in radius is
C = 2π r = 2π ( 6 ) = 12π in , hence the unit
conversion ratio
20 min ×
1 revolution
.
12π in
1 revolution
12π in
×
1 min
1 revolution
= 20 min ×
1 revolution
12π in
×
1 min
1 revolution
= 240π in
MATH 1330 Precalculus
175
Odd-Numbered Answers to Exercise Set 4.3: Unit Circle Trigonometry
1.
y
(a)
30
x
y
(b)
5.
(a)
y
x
90
x
−135
y
(c)
y
(b)
300
x
x
−π
3.
y
(a)
y
(c)
π
3
x
x
450
y
(b)
7.
y
(a)
7π
4
x
x
−240
(c)
176
y
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 4.3: Unit Circle Trigonometry
y
(b)
(b) The reference angle is 30 , shown in blue below.
y
x
13π
6
−30
x
30
y
(c)
(c) The reference angle is 60 , shown in blue below.
y
x
−510
60
60
x
9.
(a) −310 , 410 , 770
(b) −560 , 160 , 520
8π 12π 22π
,
,
5
5
5
3π π 5π
(b) −
, ,
2 2 2
11. (a) −
19. (a) The reference angle is
π
6
, shown in blue below.
y
13. 0 , 90 , 180 , 270
7π
6
15.
7π 9π 11π 13π
,
,
,
4
4
4
4
x
π
6
17. (a) The reference angle is 60 , shown in blue below.
y
(b) The reference angle is
π
3
, shown in blue below.
y
240
x
60
5π
3
x
π
3
Continued in the next column…
MATH 1330 Precalculus
Continued on the next page…
177
Odd-Numbered Answers to Exercise Set 4.3: Unit Circle Trigonometry
π
(c) The reference angle is
4
y
23.
, shown in blue below.
π
y
2
1
−
π
7π
4
π
4
0
1
-1
x
x
2π
-1
3π
2
21. (a) The reference angle is 60 , shown in blue below.
y
25.
y
2π
3
840
π
3
1
60
π
x
0
2π
1
-1
x
-1
4π
3
π
(b) The reference angle is
6
, shown in blue below.
5π
3
y
27.
y
π
−
37π
6
3π
4
2π
3
2
π
3
1
π
4
π
5π
6
x
π
6
6
π
1
-1
0
2π
x
(c) The reference angle is 60 , shown in blue below.
−660
11π
6
7π
6
y
5π
4
60
x
4π
3
-1
3π
2
5π
3
7π
4
29. II
31. III
33. IV
35. =
178
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 4.3: Unit Circle Trigonometry
51. The terminal side of the angle intersects the unit
circle at the point ( 0, − 1) .
37. <
39. >
41. sin (θ ) =
2
3
csc (θ ) =
3
2
cos (θ ) =
5
3
sec (θ ) =
3 5
5
tan (θ ) =
2 5
5
cot (θ ) =
5
2
5π
sin −
= −1
2
5π
cos −
=0
2
5π
csc −
2
5π
sec −
2
5π
tan −
is undefined
2
= −1
is undefined
5π
cot −
=0
2
53. (a) cos ( 300 ) = cos ( 60 )
(
)
(
)
( )
(b) tan 135 = − tan 45
3
43. sin (θ ) = −
5
4
cos (θ ) =
5
tan (θ ) = −
3
4
1
5
tan (θ ) = −2 6
cot (θ ) = −
2π
(b) sec −
3
4
3
sec (θ ) = −5
6
12
(
)
( )
cos ( 90 ) = 0
tan ( 90 ) is undefined
( )
sec ( 90 ) is undefined
cot ( 90 ) = 0
csc 90 = 1
2
2
csc 45 = 2
( ) 22
tan ( 45 ) = 1
sec 45 = 2
( )
( )
cot ( 45 ) = 1
cos 45 =
49. The terminal side of the angle intersects the unit
circle at the point (1, 0 ) .
sin ( −2π ) = 0
csc ( −2π ) is undefined
cos ( −2π ) = 1
sec ( −2π ) = 1
tan ( −2π ) = 0
cot ( −2π ) is undefined
MATH 1330 Precalculus
( )
59. The terminal side of the angle intersects the unit
2 2
circle at the point
,
.
2 2
( )
47. The terminal side of the angle intersects the unit
circle at the point ( 0, 1) .
π
= csc
6
(b) cot −460 = cot 80
sin 45 =
sin 90 = 1
π
= − sec
3
25π
57. (a) csc
6
5 6
csc (θ ) =
12
cot (θ ) = −
( )
55. (a) sin 140 = sin 40
5
sec (θ ) =
4
2 6
45. sin (θ ) =
5
cos (θ ) = −
5
csc (θ ) = −
3
61. The terminal side of the angle intersects the unit
3 1
,− .
circle at the point −
2
2
(
)
(
)
(
)
sin 210 = −
cos 210 = −
tan 210 =
1
2
3
2
3
3
(
)
(
)
(
)
csc 210 = −2
sec 210 = −
2 3
3
cot 210 = 3
179
Odd-Numbered Answers to Exercise Set 4.3: Unit Circle Trigonometry
63. The terminal side of the angle intersects the unit
1
3
circle at the point , −
.
2
2
5π
sin
3
3
=−
2
5π
csc
3
2 3
=−
3
67. (a) −
3
2
(b) −1
69. (a)
2
2
(b)
3
3
(b) −
2 3
3
2
2
5π 1
cos
=
3 2
5π
sec
=2
3
71. (a)
5π
tan
=− 3
3
3
5π
cot
=−
3
3
73. (a) −2
(b) −1
75. (a) 0
(b) −
77. (a) Undefined
(b) −
65. (a)
10
30
60o
6
o
30
5 3
Diagram 1
(b) sin ( 30 ) =
60o
5
5 1
=
10 2
3
o
5 3
3
cos 30 =
=
10
2
( )
5
3
tan 30 =
=
3
5 3
( )
( )
sin 60 =
79. (a)
2 3
3
5 3
3
=
10
2
5 1
cos 60 =
=
10 2
( )
5 3
tan 60 =
= 3
5
( )
sin 60 =
cos 30 =
( )
3 3
3
=
6
2
cos 60 =
( )
3
3
=
3
3 3
tan 60 =
1
sin 60 =
tan 30 =
(d) sin ( 30 ) = 2
( )
3 3
3
=
6
2
( )
3 1
=
6 2
( )
3 3
= 3
3
( )
3
2
( )
1
2
cos 30 =
( )
3
2
cos 60 =
( )
3
3
tan 60 = 3
tan 30 =
3
3
(b) −1
(b)
2
83. (a) 0.6018
(b) −0.7813
85. (a) −5.2408
(b) 2.6051
87. (a) 0.8090
(b) 1.0257
89. (a) −4.6373
(b) −1.0101
3 1
=
6 2
(c) sin ( 30 ) =
1
2
3 3
Diagram 2
81. (a) −
2
2
( )
(e) The methods in parts (b) – (d) all yield the same
results.
180
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 4.4:
Trigonometric Expressions and Identities
1.
27. sin (θ )
cos 2 (θ ) + sin 2 (θ ) = 1
cos 2 (θ )
2
cos (θ )
sin 2 (θ )
+
2
cos (θ )
=
29. sec 2 ( x )
1
2
cos (θ )
31. sec ( x )
1 + tan 2 (θ ) = sec2 (θ )
33. cos 2 ( x )
3.
sin (θ ) = −
12
12
; tan (θ ) = −
13
5
5.
sin (θ ) = −
1
9
cos (θ ) = −
4 5
9
( given )
35. tan 2 (θ )
csc (θ ) = −9
37. cos 2 (θ )
39. 1
tan (θ ) =
7.
5
20
sec (θ ) = −
9 5
20
cot (θ ) = 4 5
43. 1
45. sin ( x )
65
4
; sin (θ ) = − ;
4
9
cot (θ ) = −
41. sec ( x )
47. tan ( x )
49. 2 tan ( x )
9.
7
6
csc (θ ) = ; sin (θ ) =
6
7
51. −1
11. sin (θ ) = −
3
5
csc (θ ) = −
5
3
cos (θ ) = −
4
5
sec (θ ) = −
5
4
tan (θ ) =
3
4
( given )
cot (θ ) =
4
3
Notes for 53-69: The symbol ∴ means, “therefore.”
Q.E.D is an abbreviation for the Latin term, “Quod
Erat Demonstrandum”—Latin for “which was to be
demonstrated”—and is frequently written when a
proof is complete.
?
53.
sec ( x ) − sin ( x ) tan ( x ) = cos ( x )
13. sin 2 (θ ) + 2sin (θ ) − 15
Left-Hand Side
sec ( x ) − sin ( x ) tan ( x )
15. csc 2 ( x ) − 6 csc ( x ) + 9
=
17.
3sin (θ ) − 7 cos (θ )
cos (θ ) sin (θ )
MATH 1330 Precalculus
cos ( x )
cos 2 ( x )
cos ( x )
= cos ( x )
23. sec (θ ) − 3 sec (θ ) − 4
25. 5cot (θ ) + 1 2 cot (θ ) − 3
1 − sin 2 ( x )
=
21. 5sin (θ ) + 7 cos (θ ) 5sin (θ ) − 7 cos (θ )
cos ( x )
sin ( x )
1
− sin ( x )
cos ( x )
cos ( x )
=
19. −3 tan (θ ) sin (θ )
Right-Hand Side
∴
sec ( x ) − sin ( x ) tan ( x ) = cos ( x )
Q.E.D.
181
Odd-Numbered Answers to Exercise Set 4.4:
Trigonometric Expressions and Identities
?
?
sec (θ ) csc (θ )
55.
= 1
tan (θ ) + cot (θ )
cot 2 ( x ) − cos 2 ( x ) = cot 2 ( x ) cos 2 ( x )
59.
Left-Hand Side
Left-Hand Side
sec (θ ) csc (θ )
Right-Hand Side
cot
1
1
1
⋅
cos (θ ) sin (θ )
=
sin (θ ) cos (θ )
+
cos (θ ) sin (θ )
=
=
1
cos (θ ) sin (θ )
cos 2 ( x ) 1 − sin 2 ( x )
sin 2 ( x )
sin 2 (θ ) + cos 2 (θ )
cos (θ ) sin (θ )
=
cos 2 ( x ) cos 2 ( x )
sin 2 ( x )
= cot 2 ( x ) cos 2 ( x )
cot 2 ( x ) − cos 2 ( x ) = cot 2 ( x ) cos 2 ( x ) Q.E.D.
∴
sec (θ ) csc (θ )
tan (θ ) + cot (θ )
cot ( x ) − tan ( x )
61.
sin ( x ) cos ( x )
Left-Hand Side
= 1
cot ( x ) − tan ( x )
Q.E.D.
?
tan ( x ) − sin ( x ) cos ( x )
=
=
sin ( x )
cos ( x )
=
sin ( x ) cos ( x )
sin ( x ) cos ( x )
=
sin ( x ) 1 − cos 2 ( x )
=
sin ( x ) sin 2 ( x )
2
cos 2 ( x )
sin
2
2
( x ) cos ( x )
=
cos ( x )
= tan ( x ) sin
( x)
1
sin ( x ) cos ( x )
⋅
cos 2 ( x ) − sin 2 ( x )
sin 2 ( x ) cos 2 ( x )
−
sin 2 ( x )
sin
2
( x ) cos2 ( x )
1
sin
2
( x)
−
1
cos 2 ( x )
= csc 2 ( x ) − sec 2 ( x )
tan ( x ) − sin ( x ) cos ( x ) = tan ( x ) sin 2 ( x )
Q.E.D.
182
cos 2 ( x ) − sin 2 ( x )
sin ( x ) cos ( x )
cos 2 ( x ) − sin 2 ( x )
cos ( x )
=
∴
=
tan ( x ) sin 2 ( x )
cos ( x )
csc 2 ( x ) − sec 2 ( x )
sin ( x ) cos ( x )
sin ( x ) − sin ( x ) cos 2 ( x )
=
Right-Hand Side
cos ( x ) sin ( x )
−
sin ( x ) cos ( x )
=
Right-Hand Side
− sin ( x ) cos ( x )
?
= csc 2 ( x ) − sec2 ( x )
sin ( x ) cos ( x )
tan ( x ) − sin ( x ) cos ( x ) = tan ( x ) sin 2 ( x )
Left-Hand Side
− cos 2 ( x )
sin 2 ( x )
=1
57.
cot 2 ( x ) cos 2 ( x )
sin 2 ( x )
=
cos (θ ) sin (θ )
1
⋅
cos (θ ) sin (θ )
1
∴
( x ) − cos ( x )
cos 2 ( x ) − cos 2 ( x ) sin 2 ( x )
1
cos (θ ) sin (θ )
=
1
cos (θ ) sin (θ )
=
Right-Hand Side
2
cos 2 ( x )
=
tan (θ ) + cot (θ )
2
∴
cot ( x ) − tan ( x )
sin ( x ) cos ( x )
= csc 2 ( x ) − sec 2 ( x )
Q.E.D.
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 4.4:
Trigonometric Expressions and Identities
?
2 cos 2 ( x ) + 5sin 2 ( x ) = 2 + 3sin 2 ( x )
63.
Left-Hand Side
2
2cos ( x ) + 5sin
2
( x)
67. Two methods of proof are shown. (Either method is
sufficient.)
Right-Hand Side
2 + 3sin 2 ( x )
Method 1: Work with the left-hand side and show
that it is equal to the right-hand side.
= 2cos 2 ( x ) + 2sin 2 ( x ) + 3sin 2 ( x )
= 2 cos ( x ) + sin
2
sin ( x )
( x ) + 3sin ( x )
1 + cos ( x )
= 2 (1) + 3sin 2 ( x )
Left-Hand Side
2
2
= 2 + 3sin
2
sin ( x )
( x)
Q.E.D.
=
sin ( x )
?
tan 4 (θ ) + tan 2 (θ ) = sec4 (θ ) − sec 2 (θ )
Left-Hand Side
sin ( x ) 1 − cos ( x )
sin 2 ( x )
=
1 − cos ( x )
sin ( x )
cos ( x )
1
−
sin ( x ) sin ( x )
= csc ( x ) − cot ( x )
= sec 2 (θ ) − 1 sec 2 (θ )
∴
2
= sec (θ ) − sec (θ )
∴ tan 4 (θ ) + tan 2 (θ )
=
=
= tan 2 (θ ) sec 2 (θ )
1 − cos ( x )
1 − cos 2 ( x )
sec 4 (θ ) − sec 2 (θ )
= tan 2 (θ ) tan 2 (θ ) + 1
4
⋅
sin ( x ) 1 − cos ( x )
Right-Hand Side
tan 4 (θ ) + tan 2 (θ )
csc ( x ) − cot ( x )
1 + cos ( x ) 1 − cos ( x )
=
65.
Right-Hand Side
1 + cos ( x )
2 cos 2 ( x ) + 5sin 2 ( x ) = 2 + 3sin 2 ( x )
∴
?
= csc ( x ) − cot ( x )
= sec4 (θ ) − sec 2 (θ )
sin ( x )
1 + cos ( x )
= csc ( x ) − cot ( x )
Q.E.D.
Continued on the next page…
Q.E.D.
MATH 1330 Precalculus
183
Odd-Numbered Answers to Exercise Set 4.4:
Trigonometric Expressions and Identities
Method 2: Work with the right-hand side and show
that it is equal to the left-hand side.
sin ( x )
1 + cos ( x )
Left-Hand Side
?
Method 1: Work with the left-hand side and show
that it is equal to the right-hand side.
= csc ( x ) − cot ( x )
=
=
=
=
=
1 − cos ( x )
sin ( x )
=
sin ( x ) 1 + cos ( x )
sin 2 ( x )
=
cos ( x )
Right-Hand Side
cos ( x )
1 − sin ( x )
1 + sin ( x )
cos ( x )
=
cos ( x ) 1 − sin ( x )
1 − sin 2 ( x )
cos ( x ) 1 − sin ( x )
cos 2 ( x )
sin ( x ) 1 + cos ( x )
=
sin ( x )
1 + cos ( x )
∴
Q.E.D.
1 − sin ( x )
cos ( x ) 1 − sin ( x )
⋅
1 + sin ( x ) 1 − sin ( x )
1 − cos 2 ( x )
= csc ( x ) − cot ( x )
?
=
Left-Hand Side
cos ( x )
1
−
sin ( x ) sin ( x )
1 − cos ( x ) 1 + cos ( x )
=
⋅
1 + cos ( x )
sin ( x )
1 + cos ( x )
1 + sin ( x )
csc ( x ) − cot ( x )
1 + cos ( x )
∴
cos ( x )
Right-Hand Side
sin ( x )
sin ( x )
69. Two methods of proof are shown. (Either method is
sufficient.)
1 − sin ( x )
cos ( x )
cos ( x )
1 + sin ( x )
=
1 − sin ( x )
cos ( x )
Q.E.D.
Continued on the next page…
184
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 4.4:
Trigonometric Expressions and Identities
Method 2: Work with the right-hand side and show
that it is equal to the left-hand side.
?
cos ( x )
1 + sin ( x )
=
Left-Hand Side
73.
1 − sin ( x )
y
7
3 5
cos ( x )
cos ( x )
1 − sin ( x )
cos ( x )
x
−2
Right-Hand Side
1 + sin ( x )
θ
α
Note: The triangle may not be drawn to scale.
1 − sin ( x ) 1 + sin ( x )
=
⋅
cos ( x )
1 + sin ( x )
=
=
=
∴
cos ( x )
1 + sin ( x )
=
3 5
7
sin (θ ) =
1 − sin 2 ( x )
cos ( x ) 1 + sin ( x )
cos 2 ( x )
2
7
sec (θ ) = −
7
2
tan (θ ) = −
3 5
2
cot (θ ) = −
2 5
15
1 + sin ( x )
1 − sin ( x )
cos ( x )
Q.E.D.
7 5
15
cos (θ ) = −
cos ( x ) 1 + sin ( x )
cos ( x )
csc (θ ) =
( given )
y
75.
θ
−5
y
71.
−2
x
α
29
θ
15
α
x
Note: The triangle may not be drawn to scale.
−1
4
Note: The triangle may not be drawn to scale.
sin (θ ) = −
1
4
cos (θ ) =
15
4
tan (θ ) = −
( given )
15
15
MATH 1330 Precalculus
csc (θ ) = −4
sec (θ ) =
4 15
15
cot (θ ) = − 15
sin (θ ) = −
2 29
29
csc (θ ) = −
29
2
cos (θ ) = −
5 29
29
sec (θ ) = −
29
5
tan (θ ) =
2
5
77. sin (θ ) = −
5
13
( given )
cot (θ ) =
csc (θ ) = −
cos (θ ) =
12
13
sec (θ ) =
tan (θ ) =
−5
12
cot (θ ) = −
5
2
13
5
13
12
12
5
185
Odd-Numbered Answers to Exercise Set 4.4:
Trigonometric Expressions and Identities
79. sin (θ ) = −
cos (θ ) = −
tan (θ ) =
186
1
3
10
10
csc (θ ) = − 10
3 10
10
sec (θ ) = −
10
3
cot (θ ) = 3
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 5.1:
Trigonometric Functions of Real Numbers
1.
2
7
sin ( t ) = −
cos ( t ) =
( given )
3 5
7
tan ( t ) = −
csc ( t ) = −
sec ( t ) =
2 5
15
7
2
9.
7 5
15
cot ( t ) = −
(a)
11. (a) −
3 5
2
sin ( t ) =
15
4
csc ( t ) =
1
4
sec ( t ) = −4
cos ( t ) = −
tan ( t ) = − 15
4 15
15
cot ( t ) = −
( given )
(b)
3
4
15
15
23. (a) 1
cos ( t ) = −
4
5
sec ( t ) = −
5
4
25. −1
( given )
(b) −
1
2
5
3
4
3
3
(b)
csc ( t ) = −
cot ( t ) =
(b)
19. (a) Undefined
3
5
3
4
(b) Undefined
(b) 2
sin ( t ) = −
tan ( t ) =
2
2
(b) −2
17. (a) 0
21. (a)
5.
3
2
13. (a) −2
15. (a)
3.
2
2
2
2
(b) −
3
3
3
4
27. − csc ( t )
29. − sec ( t )
31. sec ( t )
7.
2 10
sin ( t ) = −
7
7 10
csc ( t ) = −
20
33. − csc ( t )
3
cos ( t ) =
7
7
sec ( t ) =
3
35. 1
tan ( t ) = −
2 10
3
( given )
cot ( t ) = −
3 10
20
37. tan ( t )
Continued on the next page…
MATH 1330 Precalculus
187
Odd-Numbered Answers to Exercise Set 5.1:
Trigonometric Functions of Real Numbers
Notes for 39-45: The symbol ∴ means, “therefore.”
Q.E.D is an abbreviation for the Latin term, “Quod
Erat Demonstrandum”—Latin for “which was to be
demonstrated”—and is frequently written when a
proof is complete.
39. Two methods of proof are shown. (Either method is
sufficient.)
Method 2: Work with the right-hand side and show
that it is equal to the left-hand side.
cos ( −t )
1 − sin ( −t )
Left-Hand Side
Method 1: Work with the left-hand side and show
that it is equal to the right-hand side.
1 − sin ( −t )
Left-Hand Side
cos ( −t )
Right-Hand Side
cos ( −t )
1 − sin ( −t )
cos ( −t )
?
= sec ( −t ) + tan ( −t )
sec ( −t ) + tan ( −t )
= sec ( t ) − tan ( t )
=
?
= sec ( −t ) + tan ( −t )
=
Right-Hand Side
=
sec ( −t ) + tan ( −t )
1 − sin ( −t )
=
=
cos ( t )
=
cos ( t )
1 + sin ( t )
1 − sin ( t )
=
cos ( t ) 1 − sin ( t )
=
⋅
1 + sin ( t ) 1 − sin ( t )
=
=
1 − sin 2 ( t )
cos ( t ) 1 − sin ( t )
=
cos 2 ( t )
=
=
1 − sin ( t )
∴
cos ( t )
cos ( −t )
1 − sin ( −t )
sin ( t )
1
−
cos ( t ) cos ( t )
1 − sin ( t )
cos ( t )
1 − sin ( t ) 1 + sin ( t )
⋅
cos ( t ) 1 + sin ( t )
1 − sin 2 ( t )
cos ( t ) 1 + sin ( t )
cos 2 ( t )
cos ( t ) 1 + sin ( t )
cos ( t )
1 + sin ( t )
cos ( −t )
1 − sin ( −t )
= sec ( −t ) + tan ( −t )
Q.E.D.
sin ( t )
1
−
cos ( t ) cos ( t )
= sec ( t ) − tan ( t )
Continued on the next page…
= sec ( −t ) + tan ( −t )
∴
cos ( −t )
1 − sin ( −t )
= sec ( −t ) + tan ( −t )
Q.E.D.
188
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 5.1:
Trigonometric Functions of Real Numbers
41. Two methods of proof are shown. (Either method is
sufficient.)
Method 2: Work with the right-hand side and show
that it is equal to the left-hand side.
sin ( t )
Method 1: Work with the left-hand side and show
that it is equal to the right-hand side.
sin ( t )
cos ( −t ) − 1
?
=
Left-Hand Side
Left-Hand Side
sin ( −t )
Right-Hand Side
sin ( t )
1 + cos ( −t )
cos ( −t ) − 1
sin ( −t )
=
=
1 + cos ( −t )
cos ( −t ) − 1
?
=
sin ( t )
1 + cos ( −t )
cos ( −t ) − 1
sin ( −t )
=
=
cos ( t ) − 1
=
sin ( t ) cos ( t ) + 1
=
cos 2 ( t ) − 1
− sin ( t ) cos ( t ) + 1
=
2
1 − cos ( t )
− sin ( t ) cos ( t ) + 1
=
2
sin ( t )
=
=
∴
cos ( t ) + 1
=
− sin ( t )
1 + cos ( −t )
∴
sin ( −t )
sin ( t )
cos ( −t ) − 1
1 + cos ( t )
− sin ( t )
1 + cos ( t ) 1 − cos ( t )
=
⋅
1 − cos ( t )
− sin ( t )
sin ( t )
cos ( t ) + 1
⋅
cos ( t ) − 1 cos ( t ) + 1
=
sin ( −t )
Right-Hand Side
sin ( t )
=
1 + cos ( −t )
=
1 + cos ( −t )
sin ( −t )
sin ( t )
cos ( −t ) − 1
=
1 − cos 2 ( t )
− sin ( t ) 1 − cos ( t )
sin 2 ( t )
− sin ( t ) 1 − cos ( t )
− sin ( t )
1 − cos ( t )
sin ( t )
cos ( t ) − 1
sin ( t )
cos ( −t ) − 1
1 + cos ( −t )
sin ( −t )
Q.E.D.
Q.E.D.
Continued on the next page…
MATH 1330 Precalculus
189
Odd-Numbered Answers to Exercise Set 5.1:
Trigonometric Functions of Real Numbers
1 + sec ( −t )
43.
csc ( −t )
Left-Hand Side
1 + sec ( −t )
?
= sin ( −t ) + tan ( −t )
45.
sin ( t − 4π )
1 + cos ( t + 2π )
Right-Hand Side
1 + cos ( t − 2π )
sin ( t + 6π )
Left-Hand Side
sin ( −t ) + tan ( −t )
sin ( t − 4π )
csc ( −t )
=
+
1 + cos ( t + 2π )
+
1 + cos ( t − 2π )
− csc ( t )
=
sin ( t )
1 + cos ( t )
+
1 + cos ( t )
sin ( t )
=
cos ( t ) + 1
cos ( t )
1
−
sin ( t )
=
=
cos ( t ) + 1 − sin ( t )
⋅
cos ( t )
1
sin 2 ( t ) + 1 + cos ( t )
sin 2 ( t ) + 1 + 2cos ( t ) + cos 2 ( t )
sin ( t ) 1 + cos ( t )
1 + sin 2 ( t ) + cos 2 ( t ) + 2cos ( t )
sin ( t ) 1 + cos ( t )
=
− sin ( t ) cos ( t ) − sin ( t )
1 + 1 + 2cos ( t )
sin ( t ) 1 + cos ( t )
cos ( t )
− sin ( t ) cos ( t )
cos ( t )
−
=
sin ( t )
cos ( t )
= − sin ( t ) − tan ( t )
2
sin ( t ) 1 + cos ( t )
=
=
2csc ( t + 4π )
1 + sec ( t )
1
cos ( t )
=
1
−
sin ( t )
=
Right-Hand Side
sin ( t + 6π )
1+
=
?
= 2 csc ( t + 4π )
=
2 + 2cos ( t )
sin ( t ) 1 + cos ( t )
2 1 + cos ( t )
sin ( t ) 1 + cos ( t )
= − sin ( −t ) + tan ( −t )
∴
1 + sec ( −t )
csc ( −t )
=
= sin ( −t ) + tan ( −t )
2
sin ( t )
= 2csc ( t )
Q.E.D.
= 2csc ( t + 4π )
∴
sin ( t − 4π )
1 + cos ( t + 2π )
+
1 + cos ( t − 2π )
sin ( t + 6π )
= 2csc ( t + 4π )
Q.E.D.
190
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 5.2:
Graphs of the Sine and Cosine Functions
1.
3.
5.
7.
(a) Period:
6
(b) Amplitude: 2
Range:
(a) Period:
2
(b) Amplitude: 5
(e) Amplitude: 1
Period:
2π
(a) Period:
π
(b) Amplitude: 4
(f) Answers vary. Following are some intervals on
which f ( x ) = sin ( x ) is increasing. (Any one
(a)
interval is sufficient.)
x
y = sin ( x )
x
y = sin ( x )
0
0
π
0
π
1
= 0.5
2
7π
6
2
≈ 0.71
2
5π
4
2
−
≈ −0.71
2
3
≈ 0.87
2
4π
3
3
−
≈ −0.87
2
1
3π
2
−1
2π
3
3
≈ 0.87
2
5π
3
3π
4
2
≈ 0.71
2
1
= 0.5
2
7π
4
6
π
4
π
3
π
2
5π
6
(b)
( −∞, ∞ )
[ −1, 1]
(d) Domain:
1.2
1.0
0.8
0.6
0.4
0.2
−0.2
−0.4
−0.6
−0.8
−1.0
−1.2
−
−
π π
− ,
2 2
3π
5π
− , −
2
2
1
= −0.5
2
11π
6
2π
0
7π 9π
,
2 2
etc…
Note: Some textbooks indicate closed intervals
rather than open intervals for the solutions
π π
above, e.g. − , , etc.
2 2
3
≈ −0.87
2
2
−
≈ −0.71
2
1
− = −0.5
2
3π 5π
,
2 2
7π
9π
− , −
2
2
9.
3π
A − , 0 ;
2
11. (a)
(b)
(c)
(d)
1
0
1
0
13. (a) x =
y
5π
D , 0
2
B ( π , − 1) ; C ( 2π , 1) ;
π
2
(b) x = 0, x = π
x
π/6
π/3
π/2
2π/3 5π/6
π
(c) x =
7π/6 4π/3 3π/2 5π/3 11π/6 2π
3π
2
15. (a) x = −π
(b) x = −2π
π
(c) x = − , x = −
2
(c)
y
3π
2
17.
y
1
1
x
−4π −7π/2 −3π −5π/2 −2π −3π/2 −π −π/2
−1
MATH 1330 Precalculus
π/2
π
3π/2 2π 5π/2 3π 7π/2 4π
x
−2π
−3π/2
−π
−π/2
π/2
π
3π/2
2π
−1
191
Odd-Numbered Answers to Exercise Set 5.2:
Graphs of the Sine and Cosine Functions
19. D
41. (a)
(b)
(c)
(d)
21. A
23. K
Period:
Amplitude:
Phase Shift:
Vertical Shift:
(e)
2π
6
0
−2
y
4
25. B
2
x
27. H
π/2
π
3π/2
2π
π/2
3π/4
π
−2
29. C
−4
−6
31. F
−8
−10
33. C
35. A
37. (a)
(b)
(c)
(d)
Period:
Amplitude:
Phase Shift:
Vertical Shift:
43. (a)
(b)
(c)
(d)
2π
4
0
0
Period:
Amplitude:
Phase Shift:
Vertical Shift:
(e)
(e)
y
2
π
1
0
0
y
4
1
2
x
x
π/2
π
3π/2
π/4
2π
−2
−1
−4
−2
−6
39. (a)
(b)
(c)
(d)
(e)
Period:
Amplitude:
Phase Shift:
Vertical Shift:
45. (a)
(b)
(c)
(d)
2π
5
0
0
Period:
Amplitude:
Phase Shift:
Vertical Shift:
(e)
6
y
2
8π
1
0
0
y
4
1
2
x
x
π/2
−2
π
3π/2
2π
2π
4π
6π
8π
−1
−4
−2
−6
192
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 5.2:
Graphs of the Sine and Cosine Functions
47. (a) Period:
(b) Amplitude:
4
3
(c) Phase Shift:
(d) Vertical Shift:
(e)
53. (a)
(b)
(c)
(d)
2
0
0
Period:
Amplitude:
Phase Shift:
Vertical Shift:
2
3
−1
0
(e)
4
y
y
3
4
/
3
2
1
x
x
1
−1
1
2
−1
−2
−3
-4 /
3
−4
49. (a)
(b)
(c)
(d)
Period:
Amplitude:
Phase Shift:
Vertical Shift:
(e)
1
55. (a) Period:
(b) Amplitude:
(c) Phase Shift:
4
1
0
−1
π
1
− 32π
(d) Vertical Shift:
0
(e)
y
y
1
x
1
2
3
4
x
−1
−3π/2
−π
−π/2
π/2
−2
−1
−3
51. (a) Period:
(b) Amplitude:
(c) Phase Shift:
(d) Vertical Shift:
(e)
2π
1
π
57. (a) Period:
π
2
0
y
2
(b) Amplitude:
(c) Phase Shift:
7
(d) Vertical Shift:
0
(e)
8
π
4
y
1
6
4
x
π/2
π
3π/2
2π
2
x
5π/2
π/8
π/4
3π/8
π/2
5π/8
3π/4
7π/8
−2
−4
−1
−6
−8
MATH 1330 Precalculus
193
Odd-Numbered Answers to Exercise Set 5.2:
Graphs of the Sine and Cosine Functions
59. (a) Period:
(b) Amplitude:
(c) Phase Shift:
63. (a)
(b)
(c)
(d)
1
3
1
3
(d) Vertical Shift:
5
Period:
Amplitude:
Phase Shift:
Vertical Shift:
2π
7
(*See note in left column)
Omit
(**See note in left column)
0
(e)
(e)
8
y
y
6
8
4
6
2
x
4
−2π
−3π/2
−π
−π/2
−2
2
−4
−6
1/3
2/3
1
4/3
x
−8
−2
61. (a)
(b)
(c)
(d)
(e)
Period:
Amplitude:
Phase Shift:
Vertical Shift:
65. (a)
(b)
(c)
(d)
4π
4
π
−2
Period:
Amplitude:
Phase Shift:
Vertical Shift:
1
2
(*See note in left column)
Omit
(**See note in left column)
0
(e)
4
3
y
y
2
2
1
x
π
2π
3π
4π
5π
x
6π
−1.25
−2
−1.00
−0.75
−0.50
−0.25
0.25
−1
−4
−2
−6
−3
−8
Notes for 63-67: For the equations
f ( x ) = A sin ( Bx − C ) + D and
Period:
Amplitude:
Phase Shift:
Vertical Shift:
2π
4
(*See note in left column)
Omit
(**See note in left column)
3
(e)
f ( x ) = A cos ( Bx − C ) + D :
* The period of the function is
67. (a)
(b)
(c)
(d)
8
y
7
6
2π
when B < 0 .
B
5
4
3
** The text only defines phase shift for B > 0 .
Therefore, the phase shift portion of the exercises
has been omitted.
2
1
−3π/2
−π
−π/2
x
π/2
π
−1
−2
194
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 5.2:
Graphs of the Sine and Cosine Functions
69. (a) Answers vary. Some solutions are shown below.
(Intermediate steps are shown in gray.)
f ( x ) = 4 sin ( 2 x )
π
f ( x ) = −4sin 2 x − = −4 sin ( 2 x − π )
2
π
f ( x ) = −4sin 2 x + = −4sin ( 2 x + π )
2
f ( x ) = 4 sin 2 ( x − π ) = 4 sin ( 2 x − 2π )
(b) Answers vary. Some solutions are shown below.
(Intermediate steps are shown in gray.)
π
π
f ( x ) = 4 cos 2 x − = 4 cos 2 x −
4
2
3π
3π
f ( x ) = 4 cos 2 x +
= 4 cos 2 x +
4
2
π
π
f ( x ) = −4 cos 2 x + = −4 cos 2 x +
4
2
3π
3π
f ( x ) = −4 cos 2 x −
= −4 cos 2 x −
4
2
71. (a) Answers vary. Some solutions are shown below.
(Intermediate steps are shown in gray.)
π
π
π
f ( x ) = 3sin ( x − 1) = 3 sin x −
2
2
2
3π
π
π
f ( x ) = 3sin ( x + 3) = 3sin x +
2
2
2
5π
π
π
f ( x ) = 3sin ( x − 5) = 3sin x −
2
2
2
π
π
π
f ( x ) = −3sin ( x + 1) = −3sin x +
2
2
2
3π
π
π
f ( x ) = −3sin ( x − 3) = −3sin x −
2
2
2
5π
π
π
f ( x ) = −3sin ( x + 5 ) = −3sin x +
2
2
2
(b) Answers vary. Some solutions are shown below.
(Intermediate steps are shown in gray.)
π
f ( x ) = −3cos x
2
π
π
f ( x ) = 3cos ( x − 2 ) = 3cos x − π
2
2
π
π
f ( x ) = −3cos ( x − 4 ) = −3cos x − 2π
2
2
π
π
f ( x ) = 3cos ( x + 2 ) = 3cos x + π
2
2
π
π
f ( x ) = −3cos ( x + 4 ) = −3cos x + 2π
2
2
73. (a) Answers vary. Some solutions are shown below.
(Intermediate steps are shown in gray.)
f ( x ) = −5sin ( 4 x ) − 2
π
f ( x ) = −5sin 4 x − − 2 = −5sin ( 4 x − 2π ) − 2
2
π
f ( x ) = 5sin 4 x − − 2 = 5sin ( 4 x − π ) − 2
4
π
f ( x ) = 5sin 4 x + − 2 = 5sin ( 4 x + π ) − 2
4
(b) Answers vary. Some solutions are shown below.
(Intermediate steps are shown in gray.)
π
π
f ( x ) = 5cos 4 x + − 2 = 5cos 4 x + − 2
8
2
3π
3π
f ( x ) = 5 cos 4 x −
− 2 = 5cos 4 x −
8
2
−2
π
π
f ( x ) = −5cos 4 x − − 2 = −5cos 4 x − − 2
8
2
3π
3π
f ( x ) = −5cos 4 x +
− 2 = −5cos 4 x +
−2
8
2
5π
f ( x ) = −5cos 4 x −
8
5π
− 2 = −5cos 4 x −
−2
2
Continued on the next page…
Continued in the next column…
MATH 1330 Precalculus
195
Odd-Numbered Answers to Exercise Set 5.2:
Graphs of the Sine and Cosine Functions
75. (a) Answers vary. Some solutions are shown below.
(Intermediate steps are shown in gray.)
π
π
π
f ( x ) = −2sin ( x + 1) + 3 = −2sin x + + 3
4
4
4
3π
π
π
f ( x ) = 2sin ( x − 3) + 3 = 2sin x −
+3
4
4
4
5π
π
π
f ( x ) = 2sin ( x + 5 ) + 3 = 2sin x +
+3
4
4
4
(b) Answers vary. Some solutions are shown below.
(Intermediate steps are shown in gray.)
3π
π
π
f ( x ) = 2cos ( x + 3) + 3 = 2cos x +
+3
4
4
4
5π
π
π
f ( x ) = 2cos ( x − 5 ) + 3 = 2cos x −
+3
4
4
4
π
π
π
f ( x ) = −2cos ( x − 1) + 3 = −2cos x − + 3
4
4
4
2π
2π t 4π
77. (a) D (t ) = 2 cos ( t − 2 ) + 4 = 2 cos
−
+4
13
13 13
(b) 3.52 meters
196
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 5.3:
Graphs of the Tangent, Cotangent, Secant, and Cosecant Functions
1.
(f) Answers vary. Following are some intervals on
which f ( x ) = tan ( x ) is increasing. (Any one
(a)
y = tan ( x )
x
x
π
0
0
2
π
7π
12
−3.73
0.58
2π
3
−1.73
1
3π
4
−1
1.73
5π
6
−0.58
3.73
11π
12
−0.27
π
0
6
π
4
π
3
5π
12
(b)
4.0
interval is sufficient.)
Undefined
0.27
12
π
y = tan ( x )
π π
− ,
2 2
π
3π
− , −
2
2
π 3π
,
2 2
3π
5π
− , −
2
2
is decreasing.
3.
(a)
x
y = sec ( x )
x
y = sec ( x )
0
1
π
−1
1.15
7π
6
−1.15
1.41
5π
4
−1.41
2
4π
3
−2
Undefined
3π
2
Undefined
2π
3
−2
5π
3
2
3π
4
−1.41
7π
4
1.41
5π
6
−1.15
11π
6
1.15
2π
1
y
π
2.0
6
1.0
π
x
π/6
π/4
π/3 5π/12 π/2 7π/12 2π/3 3π/4 5π/6 11π/12 π
4
−1.0
π
−2.0
3
−3.0
π
−4.0
2
(c)
4.0
y
3.0
2.0
1.0
x
−2π
−3π/2
−π
−π/2
π/2
π
etc…
There are no intervals on which f ( x ) = tan ( x )
3.0
π/12
3π 5π
,
2 2
3π/2
2π
−1.0
−2.0
−3.0
−4.0
(b)
(d) Domain:
Range:
π
3π
5π
, ±
,...
x x ≠ ± , ±
2
2
2
( −∞, ∞ )
2.5
y
2.0
1.5
1.0
0.5
x
π/6
π/3
π/2
2π/3 5π/6
π
7π/6 4π/3 3π/2 5π/3 11π/6 2π
−0.5
−1.0
(e) Period:
π
−1.5
−2.0
−2.5
−3.0
Continued in the next column…
Continued in the next column…
MATH 1330 Precalculus
197
Odd-Numbered Answers to Exercise Set 5.3:
Graphs of the Tangent, Cotangent, Secant, and Cosecant Functions
(c)
4.0
(g) Answers vary. Following are some intervals on
which f ( x ) = sec ( x ) is increasing. (Any two of
y
3.0
2.0
these intervals are sufficient.)
1.0
x
−2π
−3π/2
−π
−π/2
π/2
π
3π/2
π
0,
2
3π
− , −π
2
2π
−1.0
−2.0
−3.0
−4.0
5.
C
7.
A
9.
G
π
,π
2
5π
2π ,
2
3π
−2π , −
2
etc…
(d) The graph of h ( x ) = cos ( x ) is shown in green
below, superimposed on the graph of
f ( x ) = sec ( x ) , shown in blue. (Asymptotes of
f ( x ) = sec ( x ) are shown in red.)
11. F
4.0
y
13. D
3.0
2.0
15. B
1.0
x
−2π
−3π/2
−π
−π/2
π/2
π
3π/2
2π
−1.0
17. C
−2.0
−3.0
19. F
−4.0
21. (a) Period: π
(b)
Notice that the graphs of f ( x ) = sec ( x ) and
8
y
6
h ( x ) = cos ( x ) coincide at the points on the
4
2
graphs where y = ±1 . The asymptotes of
−3π/4
f ( x ) = sec ( x ) occur at the same x-values as the
−π/2
x
−π/4
π/4
π/2
3π/4
π/4
π/2
3π/4
−2
−4
x-intercepts of h ( x ) = cos ( x ) .
−6
−8
−10
When the graph of f ( x ) = sec ( x ) is desired,
one can begin with the graph of h ( x ) = cos ( x ) ,
and then use the key points mentioned above to
create the graph of f ( x ) = sec ( x ) .
23. (a) Period: π
(b)
8
(e) Domain:
Range:
π
3π
5π
, ±
,...
x x ≠ ± , ±
2
2
2
( −∞, 1] ∪ [1, ∞ )
y
6
4
2
−3π/4
−π/2
−π/4
x
−2
−4
(f) Period:
2π
−6
−8
−10
Continued in the next column…
198
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 5.3:
Graphs of the Tangent, Cotangent, Secant, and Cosecant Functions
25. (a) Period: π3
33. (a) Period: 2π
(b)
(b)
y
2
y
x
4
−π/2
π/2
π
3π/2
2π
−2
2
−4
x
−π/12
π/12
π/6
π/4
π/3
−6
5π/12
−8
−2
−10
−4
−12
27. (a) Period: 4π
35. (a) Period: 1
(b)
8
(b)
y
y
6
6
4
4
2
2
x
−5π
−4π
−3π
−2π
−π
x
π
−0.75
−2
−0.50
−0.25
0.25
0.50
0.75
−2
−4
−4
37. (a) The graph of f ( x ) = sin ( x ) is shown in green
1
4
29. (a) Period:
(b)
6
below, superimposed on the graph of
g ( x ) = csc ( x ) , shown in blue. (Asymptotes of
y
g ( x ) = csc ( x ) are shown in red.)
4
2
-1/4
-3/16
-1/8
-1/16
−2
1/16
1/8
3/16
1/4
4.0
x
y
3.0
−4
−6
2.0
−8
1.0
x
−10
−2π
−12
−3π/2
−π
−π/2
π/2
π
3π/2
2π
−1.0
−14
−2.0
−3.0
−4.0
31. (a) Period: π
(b)
3
(b) The x-intercepts of f ( x ) = sin ( x ) occur at
y
x = −2π , − π , 0, π , 2π .
2
1
x
−2π/3 −π/2
−π/3
−π/6
π/6
π/3
π/2
2π/3
5π/6
π
7π/6
(c) The asymptotes of g ( x ) = csc ( x ) are
−1
x = −2π , x = −π , x = 0, x = π , x = 2π . The
−2
asymptotes of g ( x ) = csc ( x ) occur at the same
−3
x-values as the x-intercepts of f ( x ) = sin ( x ) .
Continued on the next page…
MATH 1330 Precalculus
199
Odd-Numbered Answers to Exercise Set 5.3:
Graphs of the Tangent, Cotangent, Secant, and Cosecant Functions
(d)
6
39. (a) Period: 2π
y
4
(b)
2
16
y
12
x
−2π/3
−π/3
π/3
8
2π/3
4
−2
x
π/2
−4
π
3π/2
2π
−4
−6
−8
−12
−16
(e)
6
y
4
2
x
−2π/3
−π/3
π/3
2π/3
41. (a) Period: 2π
(b)
15
−2
y
12
−4
9
−6
6
3
x
π/2
(f)
6
y
−π
2π
3π
4π
−6
2
−3π/2
3π/2
−3
4
−2π
π
x
−π/2
π/2
π
3π/2
2π
−2
−4
43. (a) Period: 4π
−6
−8
(b)
−10
−12
4
y
3
−14
2
1
x
(g)
6
y
π
−1
4
2
−2π
−3π/2
−π
−π/2
2π
−2
x
π/2
π
3π/2
−3
2π
−2
−4
−4
−6
−8
−10
−12
−14
45. (a) Period:
(b)
12
2π
3
y
9
6
3
x
π/6
π/3
π/2
2π/3
−3
−6
−9
−12
−15
200
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 5.3:
Graphs of the Tangent, Cotangent, Secant, and Cosecant Functions
47. (a) Period: 1
(b)
55. (a) Period:
y
7.0
6.0
5.0
4.0
3.0
2.0
1.0
2π
3
(b)
6.0
y
4.0
2.0
x
x
0.25
−1.0
−2.0
−3.0
−4.0
−5.0
−6.0
0.50
0.75
−π/3
1.00
−π/6
π/6
π/3
−2.0
−4.0
−6.0
49. (a) Period: 6
57. (a) Period: 6π
(b)
14.0
y
(b)
12.0
10.0
10.0
y
8.0
8.0
6.0
6.0
4.0
2.0
−6.0
−4.5
−3.0
4.0
x
2.0
−1.5
x
−2.0
π
−4.0
51. (a) Period: 2π
6.0
3π
4π
5π
6π
7π
8π
−2.0
−6.0
(b)
2π
59. (a) Answers vary. Some solutions are shown below.
(Intermediate steps are shown in gray.)
y
4.0
f ( x ) = 5 tan ( x ) + 3
2.0
x
π/2
π
3π/2
2π
5π/2
3π
f ( x ) = 5 tan ( x − π ) + 3
f ( x ) = 5 tan ( x + π ) + 3
−2.0
−4.0
−6.0
(b) Answers vary. Some solutions are shown below.
(Intermediate steps are shown in gray.)
−8.0
−10.0
π
f ( x ) = −5cot x − + 3
2
π
f ( x ) = −5cot x + + 3
2
3
π
f ( x ) = −5cot x +
+3
2
53. (a) Period: 2
(b)
6.0
y
4.0
2.0
x
−0.5
0.5
−2.0
1.0
1.5
Continued on the next page…
−4.0
−6.0
MATH 1330 Precalculus
201
Odd-Numbered Answers to Exercise Set 5.3:
Graphs of the Tangent, Cotangent, Secant, and Cosecant Functions
61. (a) Answers vary. Some solutions are shown below.
(Intermediate steps are shown in gray.)
65. (a) Answers vary. Some solutions are shown below.
(Intermediate steps are shown in gray.)
π
π
f ( x ) = −3 tan 2 x + − 3 = −3 tan 2 x + − 3
8
4
3π
3π
f ( x ) = −3 tan 2 x −
− 3 = −3 tan 2 x −
−3
8
4
5π
5π
f ( x ) = −3 tan 2 x +
− 3 = −3 tan 2 x +
8
4
−3
(b) Answers vary. Some solutions are shown below.
(Intermediate steps are shown in gray.)
π
π
f ( x ) = 3cot 2 x − − 3 = 3cot 2 x − − 3
4
8
3π
3π
f ( x ) = 3 cot 2 x +
− 3 = 3cot 2 x +
−3
4
8
π
π
π
f ( x ) = 4 csc ( x + 1) + 3 = 4 csc x + + 3
4
4
4
5π
π
π
f ( x ) = −4 csc ( x + 5 ) + 3 = −4 csc x +
+3
4
4
4
(b) Answers vary. Some solutions are shown below.
(Intermediate steps are shown in gray.)
π
π
π
f ( x ) = 4sec ( x − 1) + 3 = 4sec x − + 3
4
4
4
3π
π
π
f ( x ) = −4sec ( x + 3) + 3 = −4sec x +
4
4
4
+3
7π
7π
f ( x ) = 3 cot 2 x +
− 3 = 3cot 2 x +
−3
4
8
63. (a) Answers vary. Some solutions are shown below.
(Intermediate steps are shown in gray.)
π
f ( x ) = 3sec x − − 1
2
π
f ( x ) = −3sec x + − 1
2
3π
f ( x ) = 3sec x +
−1
2
3π
f ( x ) = −3sec x −
−1
2
(b) Answers vary. Some solutions are shown below.
(Intermediate steps are shown in gray.)
f ( x ) = 3csc ( x ) − 1
f ( x ) = −3csc ( x − π ) − 1
f ( x ) = −3csc ( x + π ) − 1
f ( x ) = 3csc ( x + 2π ) − 1
202
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 5.4: Inverse Trigonometric Functions
1.
(e) The inverse relation in part (d) is not a function.
The graph does not pass the vertical line test.
(a)
x
y = cos ( x )
0
1
π
x
(f) i.
0
−
2
π
−1
3π
2
0
2π
1
y = cos ( x )
π
0
2π
3π/2
2
π
−1
−π
y
π/2
3π
2
0
−2π
1
−
x
−1.2 −1.0 −0.8 −0.6 −0.4 −0.2
0.2
0.4
0.6
0.8
1.0
1.2
−π/2
−π
−3π/2
(b)
y
−2π
1
The above graph represents a function.
x
−2π
−3π/2
−π
−π/2
π/2
π
3π/2
2π
ii.
2π
y
3π/2
−1
π
π/2
(c)
x
x = cos ( y )
y
1
0
x = cos ( y )
−1.2 −1.0 −0.8 −0.6 −0.4 −0.2
y
0.2
0.4
0.6
0.8
1.0
1.2
−π/2
0
−π
π
0
−
2
π
−3π/2
2
−2π
−1
π
−1
0
3π
2
0
−
1
2π
1
−2π
−π
The above graph does not represent a
function.
3π
2
iii.
2π
y
3π/2
(d)
2π
y
π
3π/2
π/2
x
π
−1.2 −1.0 −0.8 −0.6 −0.4 −0.2
π/2
0.2
0.4
0.6
0.8
1.0
1.2
−π/2
x
−π
−1.2 −1.0 −0.8 −0.6 −0.4 −0.2
0.2
−π/2
−π
−3π/2
0.4
0.6
0.8
1.0
1.2
−3π/2
−2π
The above graph represents a function.
−2π
Continued in the next column…
MATH 1330 Precalculus
Continued on the next page…
203
Odd-Numbered Answers to Exercise Set 5.4: Inverse Trigonometric Functions
iv.
2π
3.
y
(a)
3π/2
x
y = tan ( x )
π
0
0
π
π/2
1
−
Undefined
−
3π
4
−1
−
π
0
x
−1.2 −1.0 −0.8 −0.6 −0.4 −0.2
0.2
0.4
0.6
0.8
1.0
4
1.2
π
−π/2
2
−π
y = tan ( x )
x
π
−1
4
π
Undefined
2
3π
4
1
−3π/2
0
−π
−2π
The above graph does not represent a
function.
(b)
1.2
y
1.0
0.8
(g) The graph of f ( x ) = cos −1 x can be seen in
0.6
0.4
0.2
subsection i of part (f), with a range of [ 0, π ] .
−π
Of the graphs in part (f), only two are functions –
one with a range of [ 0, π ] and the other with a
−3π/4
−π/2
−π/4
−0.2
encompasses angles with positive measure rather
than negative measure. In terms of the unit
circle, [ 0, π ] encompasses first quadrant angles
−1.2
(c)
x = tan ( y )
y
0
0
and fourth quadrant angles.
1
y
−
Cosine
π
0
1
-1
−
x = tan ( y )
π
−1
−
Undefined
−
4
π
2
y
π
4
π
2
−1
3π
4
1
3π
−
4
0
π
0
−π
x
+
-1
−
ii.
Undefined
+
1
π
−1.0
(along with second quadrant angles as well),
while the range of [ −π , 0] encompasses third
2
3π/4
−0.6
−0.8
π
π/2
−0.4
range of [ −π , 0] . The range of [ 0, π ]
(h) i.
x
π/4
π
2
Quadrants I and II; Quadrants III and IV
(d)
π
y
3π/4
π/2
π/4
x
iii. Quadrants I and II; the range of
f ( x ) = cos −1 ( x ) is [ 0, π ] .
−1.2 −1.0 −0.8 −0.6 −0.4 −0.2
0.2
0.4
0.6
0.8
1.0
1.2
−π/4
−π/2
−3π/4
−π
204
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 5.4: Inverse Trigonometric Functions
(e) The inverse relation in part (d) is not a function.
The graph does not pass the vertical line test.
iv.
π
y
3π/4
(f) i.
π
π/2
y
π/4
3π/4
x
π/2
−1.2 −1.0 −0.8 −0.6 −0.4 −0.2
π/4
0.2
0.4
0.6
0.8
1.0
1.2
−π/4
x
−1.2 −1.0 −0.8 −0.6 −0.4 −0.2
0.2
0.4
0.6
0.8
1.0
−π/2
1.2
−π/4
−3π/4
−π/2
−π
The above graph represents a function.
−3π/4
−π
The above graph represents a function.
ii.
π
y
3π/4
π/2
π/4
x
−1.2 −1.0 −0.8 −0.6 −0.4 −0.2
0.2
0.4
0.6
0.8
1.0
1.2
−π/4
−π/2
(g) The graph of f ( x ) = tan −1 x can be seen in
subsection iv of part (f), with a range of
π π
− , . Of the graphs in part (f), the
2 2
following represent functions:
π π
i. 0, ∪ , π
2 2
π π
ii. −π , − ∪ − , 0
2 2
π π
iv. − ,
2 2
−3π/4
−π
The above graph represents a function.
iii.
π
y
3π/4
π/2
π/4
Unlike the other two functions in part (f),
π π
− , can be represented by a single
2 2
interval. In terms of the unit circle, the interval
π π
− , also encompasses first quadrant
2 2
angles (along with second quadrant angles as
well).
y
(h) i.
x
−1.2 −1.0 −0.8 −0.6 −0.4 −0.2
0.2
0.4
0.6
0.8
1.0
−3π/4
Continued in the next column…
MATH 1330 Precalculus
1
−
+
π
0
+
x
1
-1
−π
The above graph does not represent a
function.
(Tangent is undefined )
2
−π/4
−π/2
Tangent
π
1.2
−
-1
−
π
2
(Tangent is undefined )
Continued on the next page…
205
Odd-Numbered Answers to Exercise Set 5.4: Inverse Trigonometric Functions
ii. Quadrants I and II; Quadrants II and III;
Quadrants III and IV; Quadrants I and IV
(d)
2π
y
3π/2
iii. Quadrants I and II; Quadrants I and IV.
π
Quadrants I and II would produce a range of
π π
0, 2 ∪ 2 , π . Quadrants I and IV
π π
would produce a range of − , . The
2 2
π π
range of f ( x ) = tan −1 ( x ) is − , .
2 2
5.
y = csc ( x )
0
Undefined
y = csc ( x )
x
x
−4
−3
−2
−1
1
1
2
−
Undefined
π
3π
2
−1
2π
Undefined
π
−3π/2
−2π
2π
y
3π/2
Undefined
−π
−
4
−π
−1
2
3
−π/2
(f) i.
π
2
(e) The inverse relation in part (d) is not a function.
The graph does not pass the vertical line test.
(a)
x
π/2
π
π/2
3π
2
1
x
−4
−3
−2
−1
1
3
4
−π/2
Undefined
−2π
2
−π
−3π/2
(b)
4
y
−2π
3
2
The above graph does not represent a
function.
1
x
−2π
−3π/2
−π
−π/2
π/2
π
3π/2
2π
−1
−2
ii.
2π
−3
y
3π/2
−4
π
π/2
(c)
x
x = csc ( y )
y
x = csc ( y )
y
−4
−3
−2
−1
1
2
3
4
−π/2
Undefined
1
0
−π
π
π
−3π/2
2
−2π
−1
2
Undefined
π
−
Undefined
−π
−1
3π
2
1
3π
−
2
Undefined
2π
Undefined
−2π
The above graph represents a function.
Continued on the next page…
Continued in the next column…
206
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 5.4: Inverse Trigonometric Functions
iii.
2π
π π
f ( x ) = csc −1 x is − , 0 ∪ 0, . These
2 2
two ranges are identical with the exception of
y = 0 , where f ( x ) = csc −1 x is undefined.
y
3π/2
π
π/2
x
−4
−3
−2
−1
1
2
3
4
y
(h) i.
−π/2
−π
−3π/2
+
1
(Cosecant is
undefined at π ) π
The above graph does not represent a
function.
2π
2
+
−2π
iv.
Cosecant
π
1
-1
−
y
(Cosecant is
undefined at 0)
0
x
−
-1
−
3π/2
π
2
π
π/2
ii.
x
−4
−3
−2
−1
1
2
3
Quadrants I and IV; Quadrants II and III
4
iii. Quadrants I and IV; the range of
π π
f ( x ) = csc −1 x is − , 0 ∪ 0, .
2 2
−π/2
−π
−3π/2
−2π
The above graph represents a function.
7.
(g) The graph of f ( x ) = csc −1 x can be seen in
subsection iv of part (f), with a range of
π π
− 2 , 0 ∪ 0, 2 . Of the graphs in part (f),
the following represent functions:
Continued in the next column…
(b) tan −1 (1) = the number (in the interval ( − π2 , π2 ) )
π
3π
ii. , π ∪ π ,
2
2
π π
iv. − , 0 ∪ 0,
2 2
In terms of the unit circle, the interval
π π
− 2 , 0 ∪ 0, 2 encompasses first quadrant
angles (along with second quadrant angles as
well).
Recall that csc ( x ) =
1
. The range of
sin ( x )
π π
g ( x ) = sin ( x ) is − , , and the range of
2 2
−1
MATH 1330 Precalculus
1
(a) cos −1 − = the number (in the interval [ 0, π ] )
2
1
whose cosine is − .
2
1 2π
Therefore, cos −1 − =
.
2 3
whose tangent is 1 .
Therefore, tan −1 (1) =
9.
π
4
.
(a) sin −1 ( −1) = the number (in the interval
− π2 , π2 ) whose sine is −1 .
Therefore, sin −1 ( −1) = −
(b) sec−1
π
2
.
( 2 ) = the number (in the interval
0, π2 ) ∪ ( π2 , π ) whose secant is
Therefore, sec−1
2.
( 2 ) = π4 .
207
Odd-Numbered Answers to Exercise Set 5.4: Inverse Trigonometric Functions
11. sin sin −1 ( 0.2 ) = the sine of the number (in the
interval − π2 , π2 ) whose sine is 0.2 .
Therefore, sin sin
−1
45. (a) −0.84
(b) Undefined
47. (a) 3.5
(b) −7
( 0.2 ) = 0.2 .
49. (a)
13. tan arctan ( 4 ) = the tangent of the number (in the
interval ( − π2 , π2 ) ) whose tangent is 4 .
4
5
53
7
51. (a)
Therefore, tan arctan ( 4 ) = 4 .
15. (a)
53. (a)
π
(b) 0
6
17. (a) Undefined
19. (a) −
21. (a)
23. (a)
π
(b) 0
(b)
4
5
13
(b) − 35
2
2
55. (a) −
3
57. (a) −2
π
59. (a)
(b)
2π
3
(b) −
2
(b)
(b) 2 6
2
2
(b)
π
2π
3
25. (a) Undefined
8
15
(b)
π
61. (a)
2
5π
6
27. (a) False
(b) True
29. (a) False
(b) False
31. (a) 1.192
(b) −0.927
(b) − 3
2π
3
(b)
5π
6
(b) −
63.
π
4
π
4
y
π
π/2
x
−0.5
33. (a) −0.608
(b) −2.462
35. (a) 1.326
(b) −0.119
37. (a) 1.212
(b) 2.246
0.5
1.0
1.5
39. (a) 1.768
1.769
(π − tan
(5) )
(π − 1.373)
2.5
3.0
3.5
−π/2
65.
−1
2.0
y
π/2
or
π/4
x
(b) 3.042
−6
41. (a) 0.430
(b) 1.875
43. (a) Undefined
(b) −0.162
−5
−4
−3
−2
−1
1
2
3
4
5
6
−π/4
−π/2
208
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 5.4: Inverse Trigonometric Functions
67.
y
3π/2
π
π/2
x
−0.5
0.5
1.0
MATH 1330 Precalculus
1.5
2.0
2.5
209
Odd-Numbered Answers to Exercise Set 6.1: Sum and Difference Formulas
1.
sin ( x )
3.
sin ( x )
37. (a)
5.
3 cos (θ )
7.
2 cos ( x )
9.
−2sin (θ )
11.
5
3
13. −
15. −
17. −
19.
21.
(b)
7π 9π 2π 3π π
=
−
=
−
12 12 12
4 6
OR
7π 10π 3π 5π π
=
−
=
−
6 4
12
12 12
(c) −
1
5
−
11
29
7π 2π 9π π 3π
=
−
= −
12 12 12 6 4
OR
7π 3π 10π π 5π
=
−
= −
12 12 12
4 6
(d) −
5π 3π 8π π 2π
=
−
= −
12 12 12 4 3
−
5π 4π 9π π 3π
=
−
= −
12 12 12 3 4
13
11
OR
2
2
39. (a)
1
2
(b)
( )
29. tan 34
(c)
3
6
(d)
25π 15π 10π 5π 5π
=
+
=
+
12
12
12
4
6
OR
35π 27π 8π 9π 2π
=
+
=
+
12
12 12
4
3
OR
35π 33π 2π 11π π
=
+
=
+
12
12 12
4
6
33. tan ( a )
π
OR
25π 21π 4π 7π π
=
+
=
+
12
12 12
4 3
27. sin ( A )
35. (a)
19π 15π 4π 5π π
=
+
=
+
12
12 12
4 3
19π 9π 10π 3π 5π
=
+
=
+
12
12 12
4
6
1
25. −
2
210
OR
5π 9π 4π 3π π
=
−
=
−
12 12 12
4 3
19π
23. cos
90
31.
5π 8π 3π 2π π
=
−
=
−
12 12 12
3 4
=
2π
12
2π 8π
=
3
12
(b)
π
4
(e)
=
3π
12
3π 9π
=
4 12
(c)
π
3
(f)
=
4π
12
(d)
43π 33π 10π 11π 5π
=
+
=
+
12
12
12
4
6
5π 10π
=
6
12
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 6.1: Sum and Difference Formulas
41. (a) −
7π
3π 4π
π π
=−
−
=− −
12
12 12
4 3
=−
?
sin ( −θ ) = − sin (θ )
43.
4π 3π
π π
−
=− −
12 12
3 4
Left-Hand Side
= sin ( 0 − θ )
= sin ( 0 ) cos (θ ) − cos ( 0 ) sin (θ )
= 0 ⋅ cos (θ ) − 1 ⋅ sin (θ )
4π 9π
π 3π
−
=− −
12 12
3 4
= − sin (θ )
OR
−
∴ sin ( −θ )
13π
10π 3π
5π π
=−
−
=−
−
12
12 12
6 4
=−
Q.E.D.
?
3π 10π
π 5π
=− −
−
12 12
4 6
Left-Hand Side
29π
2π 27π
π 9π
=−
−
=− −
12
12 12
6 4
Right-Hand Side
− tan (θ )
= tan ( 0 − θ )
=
9π π
27π 2π
=−
−
=−
−
12 12
4 6
tan ( 0 ) − tan (θ )
1 + tan ( 0 ) tan (θ )
=
OR
−
= − sin (θ )
tan ( −θ ) = − tan (θ )
45.
tan ( −θ )
(c) −
− sin (θ )
sin ( −θ )
13π
9π 4π
3π π
(b) −
=−
−
=−
−
12
12 12
4 3
=−
Right-Hand Side
0 − tan (θ )
1 + 0 ⋅ tan (θ )
= − tan (θ )
29π
8π 21π
2π 7π
=−
−
=−
−
12
12 12
3
4
∴
tan ( −θ ) = − tan (θ )
Q.E.D.
21π 8π
7π 2π
=−
−
=−
−
12 12
4
3
(d) −
47.
6+ 2
4
49.
2− 6
4
41π
33π 8π
11π 2π
=−
−
=−
−
12
12 12
4
3
8π 33π
2π 11π
=−
−
=−
−
12 12
3
4
51. −2 − 3
MATH 1330 Precalculus
53.
6+ 2
4
55.
2− 6
4
57.
2− 6
4
211
Odd-Numbered Answers to Exercise Set 6.1: Sum and Difference Formulas
65. The following diagrams, though not required, may be
helpful in setting up the problems:
59. −2 + 3
61. 2 − 3
y
63. The following diagrams, though not required, may be
helpful in setting up the problems:
y
sin (α ) = −
2
sin (α ) =
5
4
α
x
−5
29
Note: The triangle may not be drawn to scale.
x
3
y
sin ( β ) = −
Note: The triangle may not be drawn to scale.
y
13
−5
sin ( β ) =
x
− 3
Note: The triangle may not be drawn to scale.
x
Answers:
(a)
−5 29 − 2 87
58
(b)
2 29 − 5 87
58
(c)
40 − 29 3
71
Answers:
56
(a) −
65
63
(b) −
65
56
33
1
2
tan ( β ) = − 3
2
Note: The triangle may not be drawn to scale.
(c) −
1
12
13
5
cos ( β ) = −
13
12
tan ( β ) = −
5
β
3
2
cos ( β ) =
β
12
2
2 29
=
29
29
5
tan (α ) = −
2
cos (α ) =
α
4
5
3
cos (α ) =
5
4
tan (α ) =
3
5
5 29
=−
29
29
67.
2
2
69.
33
56
71. cos ( A − B )
73. sin 2 ( A ) − sin 2 ( B ) OR
cos 2 ( B ) − cos 2 ( A )
212
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 6.1: Sum and Difference Formulas
?
sin ( x − y ) + sin ( x + y ) = 2 sin ( x ) cos ( y )
75.
Left-Hand Side
79.
sin ( x − y )
sin ( x + y )
?
=
tan ( x ) − tan ( y )
tan ( x ) + tan ( y )
Right-Hand Side
Left-Hand Side
2sin ( x ) cos ( y )
sin ( x − y ) + sin ( x + y )
= sin ( x ) cos ( y ) − cos ( x ) sin ( y ) +
Right-Hand Side
sin ( x − y )
tan ( x ) − tan ( y )
sin ( x + y )
tan ( x ) + tan ( y )
=
sin ( x ) sin ( y )
−
cos( x ) cos ( y )
sin ( x ) sin ( y )
+
cos ( x ) cos ( y )
=
sin ( x ) cos( y ) − cos( x ) sin ( y )
cos( x ) cos( y )
sin ( x ) cos ( y ) + cos ( x ) sin ( y )
cos( x ) cos( y )
=
sin ( x − y )
cos( x ) cos( y )
sin ( x + y )
cos( x ) cos( y )
sin ( x ) cos ( y ) + cos ( x ) sin ( y )
= 2sin ( x ) cos ( y )
∴ sin ( x − y ) + sin ( x + y )
= 2 sin ( x ) cos ( y )
Q.E.D.
cos ( x − y ) + cos ( x + y )
77.
sin ( x ) sin ( y )
?
= 2 cot ( x ) cot ( y )
=
Left-Hand Side
cos ( x − y ) + cos ( x + y )
Right-Hand Side
2cot ( x ) cot ( y )
=
sin ( x ) sin ( y )
=
cos ( x ) cos ( y ) + sin ( x ) sin ( y )
sin ( x ) sin ( y )
∴
+
sin ( x − y )
sin ( x + y )
=
sin ( x − y ) cos ( x ) cos ( y )
cos ( x ) cos ( y ) sin ( x + y )
sin ( x − y )
sin ( x + y )
tan ( x ) − tan ( y )
tan ( x ) + tan ( y )
Q.E.D.
cos ( x ) cos ( y ) − sin ( x ) sin ( y )
sin ( x ) sin ( y )
=
=
2cos ( x ) cos ( y )
sin ( x ) sin ( y )
2cos ( x ) cos ( y )
⋅
sin ( x ) sin ( y )
81. (a) sin ( A ) =
a
c
(b) cos ( B ) =
a
c
(c) sin ( A ) = cos ( B )
= 2cot ( x ) cot ( y )
(d) A = 90 − B
∴
cos ( x − y ) + cos ( x + y )
sin ( x ) sin ( y )
MATH 1330 Precalculus
= 2cot ( x ) cot ( y )
B = 90 − A
(
cos ( A ) = sin ( 90
)
− A)
(e) sin ( A ) = cos 90 − A
213
Odd-Numbered Answers to Exercise Set 6.1: Sum and Difference Formulas
83. (a) sec ( A ) =
c
b
(b) csc ( B ) =
c
b
(c) sec ( A ) = csc ( B )
(d) A = 90 − B
B = 90 − A
(
csc ( A) = sec ( 90
)
− A)
(e) sec ( A ) = csc 90 − A
85. x = 15
87. x =
π
10
89. x = 18
91. 1
93. sin ( x )
95. cot (θ )
97. −1
99. −1
214
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 6.2:
Double-Angle and Half-Angle Formulas
1.
(a)
3
2
(b) 1
(d)
y
(c) No
11. (a) −
g(x)
13.
2
f(x)
4 21
25
(b)
17
25
(c)
−4 21
17
1
2
1
x
π/2
π
3π/2
2π
( )
15. cos 68
−1
−2
17.
3
3
19.
2
−3
(e) No
3.
(a)
(b)
3
2 3
3
(d)
21. −
(c) No
6 y
4
f(x)
23.
g(x)
2
2
3
2
2
x
−π/2
−π/4
π/4
π/2
( )
25. − tan 82
−2
−4
27. cos ( β )
−6
Notes:
Only one ‘branch’ of each graph is shown. In the
portion of each graph shown,
f ( x ) has asymptotes at x = ± π4 , and
g ( x ) has asymptotes at x = ± π2 .
31. (a) Quadrant IV; Positive
(b) Quadrant III; Negative
1 − cos ( s )
s
33. (a) sin = ±
2
2
(e) No
5.
29. (a) Quadrant II; Negative
(b) Quadrant I; Positive
1 + cos ( s )
s
(b) cos = ±
2
2
cos ( 2θ ) = cos (θ + θ )
= cos (θ ) cos (θ ) − sin (θ ) sin (θ )
= cos 2 (θ ) − sin 2 (θ )
7.
s
(c) tan =
2
(a) cos 2 (θ ) = 1 − sin 2 (θ )
(b) cos ( 2θ ) = cos 2 (θ ) − sin 2 (θ )
1 − cos ( s )
2
=±
1 + cos ( s )
±
2
±
= 1 − sin 2 (θ ) − sin 2 (θ )
=±
= 1 − 2 sin 2 (θ )
9.
(a) −
120
169
(b)
119
169
MATH 1330 Precalculus
(c) −
120
119
=±
1 − cos ( s )
2
⋅
1 − cos ( s )
2
1 + cos ( s )
2
2
1 + cos ( s )
1 − cos ( s )
1 + cos ( s )
215
Odd-Numbered Answers to Exercise Set 6.2:
Double-Angle and Half-Angle Formulas
( )
=±
=±
1 − cos ( s )
⋅
1 − cos ( s )
1 + cos ( s ) 1 − cos ( s )
1 − cos ( s )
sin
2
=±
=±
1 − cos ( s )
1 − cos
2
2 3
2 1
6
2
⋅
−
⋅ =
−
2 2
2 2
2
2
6− 2
=
2
=
(s)
1 − cos ( s )
(s)
(
)
= cos ( 45 ) cos ( 30 ) − sin ( 45 ) sin ( 30 )
37. (a) cos 75 = cos 45 + 30
1 − cos ( s )
s
35. (a) tan = ±
1 + cos ( s )
2
sin ( s )
Other numbers may be chosen that have a sum or
(
1 − cos ( s )
s
(c) tan = ±
1 + cos ( s )
2
the end result is the same.
(
1 + cos 150
( )
(b) cos 75 = +
2
sin ( s )
s
tan =
2 1 + cos ( s )
=
2− 3
2
=
2
s 1 − cos ( s )
tan =
sin ( s )
2
=
2− 3
2
s 1 − cos ( s )
Of the formulas above, tan =
is
sin ( s )
2
generally easiest to use, since the denominator
contains only one term. This is particularly true
when the denominator needs to be rationalized.
In the case where the denominator does not
sin ( s )
s
contain a radical sign, tan =
is
2 1 + cos ( s )
(c)
39. (a)
(b) −
is generally the most difficult to use. One must
always choose whether or not the answer is
positive or negative, as well as rationalizing the
denominator. (One advantage of this formula is
that it does not require the user to find sin ( s )
)=
1+
( − 23 )
2
2− 3
4
6− 2
2− 3
≈ 0.26 ;
≈ 0.26 .
4
2
They are the same.
equally easy to use.
1 − cos ( s )
s
Of the formulas above, tan = ±
1 + cos ( s )
2
)
difference of 75 , such as cos 105 − 30 , but
s 1 − cos ( s )
(b) tan =
sin ( s )
2
2+ 2
2
2− 2
2
(c)
2 −1
(d)
2 −1
41. (a) −
2− 3
2
(b) −
2+ 3
2
before computing the answer.)
(c) 2 − 3
216
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 6.2:
Double-Angle and Half-Angle Formulas
43. (a) −
2+ 2
2
(b) −
2− 2
2
(c)
1 − cos ( 2 x )
51.
sin ( 2 x )
Left-Hand Side
tan ( x )
sin ( 2 x )
=
2+ 3
2
1 − 1 − 2sin 2 ( x )
2sin ( x ) cos ( x )
1 − 1 + 2sin 2 ( x )
=
2− 3
2
(b)
Right-Hand Side
1 − cos ( 2 x )
2 +1
45. (a) −
?
= tan ( x )
2sin ( x ) cos ( x )
=
(c) −2 − 3
2sin 2 ( x )
2sin ( x ) cos ( x )
=
47. (a) IV
sin ( x )
cos ( x )
= tan ( x )
3π θ
(b)
< <π
4 2
∴ 1 − cos ( 2 x )
sin ( 2 x )
(c) II
= tan ( x )
Q.E.D.
(d) Positive
(e) Negative
?
(f)
2
3
53.
(g) −
7
3
(h) −
14
7
49. (a)
14
4
(b)
2
4
(c)
7
MATH 1330 Precalculus
1 − 2 sin ( x ) 1 + 2 sin ( x ) = cos ( 2 x )
Left-Hand Side
1 − 2 sin ( x ) 1 + 2 sin ( x )
Right-Hand Side
cos ( 2 x )
= 1 − 2sin 2 ( x )
= cos ( 2 x )
∴
1 − 2 sin ( x ) 1 + 2 sin ( x ) = cos ( 2 x )
Q.E.D.
Continued on the next page…
217
Odd-Numbered Answers to Exercise Set 6.2:
Double-Angle and Half-Angle Formulas
?
csc ( x ) − cot ( x )
55.
Left-Hand Side
csc ( x ) − cot ( x )
=
= tan
2
x
Right-Hand Side
x
tan
2
cos ( x )
1
−
sin ( x ) sin ( x )
=
1 − cos ( x )
*
sin ( x )
1 − cos ( x ) 1 + cos ( x )
=
⋅
sin ( x )
1 + cos ( x )
=
=
1 − cos 2 ( x )
sin ( x ) 1 + cos ( x )
sin 2 ( x )
sin ( x ) 1 + cos ( x )
=
sin ( x )
1 + cos ( x )
x
= tan
2
x
csc ( x ) − cot ( x ) = tan
2
Q.E.D.
*Note about the proof above: Exercise 35
x 1 − cos ( x )
establishes the identity tan =
. If
sin ( x )
2
this identity is used, several steps can be omitted
from the proof above. In the proof above, the
sin ( x )
x
identity from the text, tan =
was
2 1 + cos ( x )
used.
218
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 6.3: Solving Trigonometric Equations
1.
No
3.
Yes
17. (a) x = 0
(b) x = 0
(c) x = 2kπ , where k is any integer.
5.
(a) x = 120 , x = 240
(b) x =
19. (a) 53.1
2π
4π
, x=
3
3
(b) x ≈ 233.1 , x ≈ 306.9
2π
4π
+ 2kπ and x =
+ 2kπ , where k is any
3
3
integer.
(c) x =
7.
9.
(a) x = 135 , x = 315
3π
7π
, x=
4
4
(c) x =
3π
+ kπ , where k is any integer.
4
(b) x ≈ 45.6 , x ≈ 134.4
(a) x = 90
(c) x =
25. (a) x ≈ 161.0 , x ≈ 341.0
(b) No solution
27. (a) x =
π
2
π
2
(b) x =
+ 2kπ , where k is any integer.
π
6
π
2
, x=
11π
6
, x=
3π
π
11π
, x= , x=
2
6
6
(c) No
(d) Part (b) is correct. Part (a) is incorrect because
dividing by cos ( x ) causes loss of the solutions
11. (a) x = 150 , x = 210
(b) x =
(b) x ≈ 207.4 , x ≈ 332.6
23. (a) x ≈ 115.4 , x ≈ 244.6
(b) x =
(b) x =
21. (a) 27.4
of cos ( x ) = 0 .
5π
7π
, x=
6
6
5π
7π
+ 2kπ and x =
+ 2kπ , where k is any
6
6
integer.
29. x = 0 , x = 180 , x = 45 , x = 135
(c) x =
33. x = 90 , x = 270 , x = 30 , x = 150 , x = 210 , x = 330
13. (a) x = 45 , x = 315
(b) x =
(c) x =
π
4
π
, x=
7π
4
+ 2kπ and x =
4
integer.
31. x = 0 , x = 180 , x = 150 , x = 330
35. x = 90 , x = 270 , x = 120 , x = 240 , x = 180
7π
+ 2kπ , where k is any
4
37. (a) 0 ≤ 2 x < 4π
(b) Let u = 2 x ;
u=
15. (a) x = 210 , x = 330
(b) x =
7π
11π
, x=
6
6
(c) x =
π
4
π
8
, u=
3π
9π
11π
, u=
, u=
4
4
4
, x=
3π
9π
11π
, x=
, x=
8
8
8
7π
11π
+ 2kπ and x =
+ 2kπ , where k is
6
6
any integer.
(c) x =
MATH 1330 Precalculus
219
Odd-Numbered Answers to Exercise Set 6.3: Solving Trigonometric Equations
39. (a) −
π
2
≤ x−
π
2
<
3π
2
69. x =
π
4π
; u=− , u=
2
3
3
π
11π
(c) x = , x =
6
6
(b) Let u = x −
π
71. x =
73. x =
π
3
, x=
5π
, x =π
3
7π
11π
π
, x=
, x=
6
6
2
π
3
, x=
2π
4π
5π
, x=
, x=
3
3
3
, x=
5π
3
, x=
5π
4
41. (a) π ≤ 3x + π < 7π
(b) Let u = 3x + π ;
75. x =
π
3
u = π , u = 2π , u = 3π , u = 4π , u = 5π , u = 6π
(c) x = 0, x =
43. x =
π
8
, x=
π
3
, x=
2π
4π
5π
, x =π, x =
, x=
3
3
3
77. x =
π
4
7π
9π
15π
, x=
, x=
8
8
8
45. x = 112.5 , x = 157.5 , x = 292.5 , x = 337.5
47. No solution
49. x = 80 , x = 100 , x = 200 , x = 220 , x = 320 , x = 340
51. x =
x=
5π
7π
17π
19π
29π
, x=
, x=
, x=
, x=
,
24
24
24
24
24
31π
41π
43π
, x=
, x=
24
24
24
53. x = 135 , x = 315
55. x = 0, x =
5π
3
57. x = 0 , x = 180
59. x = 330
61. x = 7.5 , x = 37.5 , x = 187.5 , x = 217.5
63. x = 0 , x = 180 , x ≈ 23.6 , x ≈ 156.4
65. x ≈ 108.4 , x ≈ 288.4 , x ≈ 26.6 , x ≈ 206.6
67. x ≈ 75.5 , x ≈ 104.4 , x ≈ 255.5 , x ≈ 284.5
220
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 7.1: Solving Right Triangles
1.
( )
(a) cos 30 =
(b)
x = 5 3;
x
;
10
( )
sin 30 =
13. x ≈ 48.43
y
10
15. x ≈ 75.96
y = 10
(c) In a 30o-60o-90o triangle, the length of the
hypotenuse (in this case, 10), is two times the
length of the shorter leg (in this case, y).
Therefore, 10 = 2 y , so y = 5 . The length of the
longer leg (in this case, x) is 3 times the length
of the shorter leg (in this case, 5). Therefore,
x=5 3.
3.
( )
(a) tan 60 =
(b) x = 7;
7 3
;
x
( )
sin 60 =
7 3
y
o
19. x ≈ 18.12
21. AC ≈ 9.18 in
23. ∠H ≈ 49.88
25. ∠G ≈ 37.87
27. ∠M = 61 (given)
y = 14
o
17. x ≈ 19.75
∠D = 90 (given)
o
(c) In a 30 -60 -90 triangle, the length of the longer
leg (in this case, 7 3 ) is 3 times the length of
the shorter leg (in this case, x). Therefore,
7 3 = x 3 , so x = 7 . The length of the
hypotenuse (in this case, y), is two times the
length of the shorter leg (in this case, 7).
Therefore, y = 2 ( 7 ) = 14 .
∠J = 29
JD = 3 (given)
MD ≈ 1.66
JM ≈ 3.43
29. ∠H = 90 (given)
∠L ≈ 22.02
5.
( )
(a) tan 45 =
(b) x = 5;
5
;
x
( )
sin 45 =
5
y
∠P ≈ 67.98
PH = 3 (given)
PL = 8 (given)
y=5 2
(c) In a 45o-45o-90o triangle, the legs are congruent,
so x = 5 . The length of the hypotenuse (in this
2 times the length of either leg (in
case, y) is
Note: The diagrams in 31-45 may not be drawn to scale.
this case, 5). Therefore, y = 5 2 .
7.
( )
(a) cos 45 =
(b) x = 7;
HL = 55 ≈ 7.42
x
7 2
;
( )
sin 45 =
y
7 2
31. (a)
14 in
14 in
14 in
14 in
y=7
o
(c) In a 45 -45o-90o triangle, the legs are congruent,
so x = y . The length of the hypotenuse (in this
case, 7 2 ) is 2 times the length of either leg
(in this case, we will use y). Therefore,
7 2 = y 2 , so y = 7 . Since the legs are
congruent, x = y = 7 .
9.
(a) 0.7431
(b) 0.2924
11. (a) 0.3640
(b) 0.7818
MATH 1330 Precalculus
9 in
4.5 in
The second diagram above may be useful, since
right triangle trigonometry can be used to find
the missing angles.
(b) The measures of the angles are:
71.3 , 71.3 , and 31.4 (or 31.5 , depending on
rounding and the method of solution).
221
Odd-Numbered Answers to Exercise Set 7.1: Solving Right Triangles
33. (a)
41. (a)
KITE
H
O
U
S
E
Ladder
8 ft
200 ft
o
72
55 o
(b) 7.6 ft
6 ft
Jeremy
35. (a)
(b) 169.8 ft
Flagpole
o
51
40 ft
43. (a)
DOUG
41
(b) 49.4 ft
C
L
I
F
F
o
60
m
37. (a)
RAM P
34
COUGAR
8 ft
o
DOCK
(b) 69.0 m
(b) 14.3 ft
45. (a)
39. (a)
Sears
Tower
MOUNTAIN
1,450
ft
33
Silas
(b) 67.5
222
o
42
o
150 m
600 ft
(b) 388.1 m
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 7.2: Area of a Triangle
8π
39. A =
− 4 3 in 2
3
1.
A = 40 cm 2
3.
A=
3 55 2
ft
2
5.
A=
25 3 2
m
2
7.
A ≈ 62.36 in 2
9.
(a) sin ( C ) =
h
b
(b) h = b sin ( C )
(c) K = 12 ab sin ( C )
11. A = 15 3 in 2
13. A = 18 2 m 2
15. A ≈ 22.53 m 2
17. A ≈ 11.48 m 2
19. A =
49 2 2
in
4
21. A = 30 3 cm 2
23. A ≈ 36.88 m 2
25. (a) x = 30 , x = 150
(b) ∠S = 30 or ∠S = 150
27. ∠R ≈ 47.73 or ∠R ≈ 132.27
29. A = 288 2 in 2
31. A = 72 3 in 2
33. 4 3 ft 2
35. 492.43 in 2
37. A ≈ 11.89 cm 2
MATH 1330 Precalculus
223
Odd-Numbered Answers to Exercise Set 7.3:
The Law of Sines and the Law of Cosines
1.
31. ∠P ≈ 27.36
b ≈ 8.76 m
∠I ≈ 95.22
3.
c = 3 6 in
5.
∠N ≈ 27.66
∠G ≈ 57.42
p = 6 cm (given)
7.
s ≈ 8.34 cm
i = 13 cm (given)
g = 11 cm (given)
9.
∠C ≈ 51.75
33. No such triangle exists.
11. No such triangle exists.
35. 653.3 cm 2
13. t = 6 2 cm
37. 277.5 in 2
15. ∠D ≈ 43.56 or ∠D ≈ 136.44
39. (a) 36.87 , 53.13
(b) 36.87 , 53.13
17. ∠C ≈ 41.62
19. No such triangle exists.
41. ∠E ≈ 14.06 , ∠Y ≈ 40.94
21. ∠A ≈ 12.56
43. AD ≈ 5.64 mm
23. ∠D = 20
∠E = 125 (given)
∠F = 35 (given)
d ≈ 5.62 ft
e ≈ 13.45 ft
f = 9.42 ft (given)
25. ∠W ≈ 82.13
∠W ≈ 41.87
∠H ≈ 69.87
∠H ≈ 110.13
∠Y = 28 (given)
∠Y = 28 (given)
w ≈ 8.44 in
h = 8 in (given)
y = 4 in (given)
w ≈ 5.69 in
h = 8 in (given)
y = 4 in (given)
27. No such triangle exists.
29. ∠P ≈ 27.42
∠E = 130 (given)
∠Z ≈ 22.58
p = 12 mm (given)
e ≈ 19.96 mm
z = 10 mm (given)
224
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 8.1: Parabolas
1.
( y − 7 ) 2 = 2 ( x + 3)
3.
( x − 4 )2 = −9 y −
5.
( y − 4)
26
9
1
= ( x − 2)
3
2
y 2 = −10 x
y=0
11. (a) Equation:
(b) Axis:
(c) Vertex:
( 0, 0 )
(d) Directrix:
x = 2 12
(e) Focus:
( −2 12 , 0 )
(f) Focal Width: 10
( −2 12 , − 5) , ( −2 12 , 5)
(g) Endpoints of focal chord:
2
7.
1
1
23
x− = y+
2
3
4
(h) Graph:
y
6
5
9.
(a) Equation:
(b) Axis:
(c) Vertex:
4
x2 = 4 y
x=0
( 0, 0 )
(d) Directrix:
y = −1
(e) Focus:
( 0, 1)
3
2
1
F
−3
−2
V
x
−1
1
2
3
4
−1
−2
(f) Focal Width: 4
(g) Endpoints of focal chord:
−3
−4
( −2, 1) , ( 2, 1)
−5
(h) Graph:
−6
−7
y
3
2
13. (a) Equation:
F
1
x
−3
−2
V
−1
−1
−2
−3
1
2
3
( x − 2 )2 = 8 ( y + 5 )
(b) Axis:
(c) Vertex:
x=2
( 2, − 5 )
(d) Directrix:
y = −7
(e) Focus:
( 2, − 3)
(f) Focal Width: 8
(g) Endpoints of focal chord:
( −2, − 3) , ( 6, − 3)
(h) Graph:
y
1
x
−6
−4
−2
2
4
6
8
10
12
−1
−2
F
−3
−4
−5
V
−6
−7
−8
−9
MATH 1330 Precalculus
225
Odd-Numbered Answers to Exercise Set 8.1: Parabolas
y 2 = 4 ( x − 3)
15. (a) Equation:
( y − 6 ) 2 = −1 ( x − 2 )
19. (a) Equation:
(b) Axis:
y=0
(b) Axis:
y=6
(c) Vertex:
( 3, 0 )
(c) Vertex:
( 2, 6 )
(d) Directrix:
x = 2 14
(e) Focus:
(1 34 , 6 )
(d) Directrix:
(e) Focus:
x=2
( 4, 0 )
(f) Focal Width: 4
(g) Endpoints of focal chord:
(h) Graph:
(f) Focal Width: 1
(g) Endpoints of focal chord:
( 4, − 2 ) , ( 4, 2 )
(1 34 , 5 12 ) , (1 34 , 6 12 )
(h) Graph:
6
y
8
5
4
y
7
3
F
6
2
1
F
V
1
2
3
4
5
x
5
6
V
7
4
−1
3
−2
−3
2
−4
1
−5
x
−6
−0.5
−7
0.5
1.0
1.5
2.0
2.5
3.0
−1
( x + 5 ) 2 = −2 ( y − 4 )
17. (a) Equation:
( x + 6 )2 = 6 ( y + 2 )
21. (a) Equation:
(b) Axis:
(c) Vertex:
( −5, 4 )
(b) Axis:
(c) Vertex:
( −6, − 2 )
(d) Directrix:
y = 4 12
(d) Directrix:
y = −3 12
(e) Focus:
( −5, 3 12 )
(e) Focus:
( −6, − 12 )
x = −5
(f) Focal Width: 2
(g) Endpoints of focal chord:
x = −6
(f) Focal Width: 6
(
−6, 3 12
), (
−4, 3 12
(h) Graph:
)
(g) Endpoints of focal chord:
( −9, − 12 ) , ( −3, − 12 )
(h) Graph:
y
y
5
V
1
4
x
F
3
−11 −10 −9
−8
−7
−6
F
2
1
−8
−7
−6
−5
−4
−3
−2
−4
−3
−2
−1
1
−1
−2
x
−9
−5
V
−3
−1
−1
−4
226
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 8.1: Parabolas
( y − 2 )2 = 8 ( x + 5)
23. (a) Equation:
( y − 2 )2 = −2 ( x − 4 )
27. (a) Equation:
(b) Axis:
y=2
(b) Axis:
y=2
(c) Vertex:
( −5, 2 )
(c) Vertex:
( 4, 2 )
(d) Directrix:
(e) Focus:
x = −7
( −3, 2 )
(d) Directrix:
x = 4 12
(e) Focus:
( 3 12 , 2 )
(f) Focal Width: 8
(g) Endpoints of focal chord:
( −3, − 2 ) , ( −3, 6 )
(h) Graph:
8
(f) Focal Width: 2
(g) Endpoints of focal chord:
( 3 12 , 1) , (3 12 , 3)
(h) Graph:
y
7
y
5
6
5
4
4
3
V
F
3
2
−8
−7
−6
−5
−4
−3
−2
F
2
1
V
x
−1
1
2
3
4
1
−1
x
−2
1
−3
2
3
4
5
−1
−4
−2
25. (a) Equation:
( x + 5)2 = −16 y
(b) Axis:
(c) Vertex:
x = −5
( −5, 0 )
(d) Directrix:
y=4
(e) Focus:
( −5, − 4 )
(f) Focal Width: 16
(g) Endpoints of focal chord:
29. (a) Equation:
( −13, − 4 ) , ( 3, − 4 )
8
3
( y + 5 )2 = ( x + 1)
(b) Axis:
y = −5
(c) Vertex:
( −1, − 5 )
(d) Directrix:
x = −1 23
(e) Focus:
( − 13 , − 5)
4
8
3
(g) Endpoints of focal chord:
( − 13 , − 6 13 ) , ( − 13 , − 3 32 )
3
(h) Graph:
(h) Graph:
5
y
(f) Focal Width:
2
V
−14
−12 −10
−8
−6
1
−4
−2
−1
1
x
2
4
6
x
−2
−1
1
−1
−2
−3
−2
−4
F
y
−5
−3
−6
−4
−7
−8
V
F
−5
−9
−10
−6
−7
−8
MATH 1330 Precalculus
227
Odd-Numbered Answers to Exercise Set 8.1: Parabolas
( x − 2 )2 = − ( y − 6 )
43.
( x − 5 )2 = 6 ( y − 6 )
(b) Axis:
(c) Vertex:
x=2
( 2, 6 )
45.
( y − 2 )2 = x + 3
(d) Directrix:
y = 6 78
(e) Focus:
(
31. (a) Equation:
7
2
2, 5 18
47. y 2 = −10 ( x − 2 12 )
)
49. (a) y = 11x − 5
(b) y = x
7
2
(g) Endpoints of focal chord:
( 14 , 5 81 ) , (3 34 , 5 81 )
(f) Focal Width:
51. (a) y = x − 3
(h) Graph:
(b) y = 7 x −
3
2
y
53. y = 3 x − 7
7
55. y = 4 x + 26
V
6
5
F
4
3
and ( 7, 32 )
57.
( 2, 7 )
59.
( −1, 1)
61.
( −3, − 32 )
63.
( 2, 1)
and ( 3, 4 )
and (1, 4 )
2
1
x
1
2
3
4
−1
33.
( y − 5)2 = 24 ( x + 2 )
35.
( x − 2 )2 = 16 y
37.
( x + 2 )2 = 12 ( y + 6 )
39.
( y + 1)
41.
( x + 4 )2 = −28 ( y − 5 )
228
2
= −7 ( x − 5 34 )
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 8.2: Ellipses
1.
3.
x2 y2
+
=1
4 25
( x − 2 )2 ( y − 4 )2
+
4
( x − 2 )2
13. (a) Equation:
9
16
( 2, 0 )
(b) Center:
=1
y2
=1
4
+
( −2, 0 ) , ( 6, 0 )
(c) Vertices of Major Axis:
8
Length of Major Axis:
5.
( x − 5)
2
+
2
( y − 3)
( x − 12 ) + ( y + 32 )
8
9.
16
( 2, − 2 ) , ( 2, 2 )
(d) Vertices of Minor Axis:
=1
3
2
7.
2
4
Length of Minor Axis:
2
(2 − 2
(e) Foci:
=1
) (
3, 0 , 2 + 2 3, 0
)
( −1.5, 0 ) , ( 5.5, 0 )
c
a
(b) elongated
(f) Eccentricity:
(a) e =
(exact)
(decimal)
3
≈ 0.9
2
e=
(g) Graph:
4
(c) circle
y
3
2
2
11. (a) Equation:
2
x
y
+
=1
9 49
1
F2
2
3
4
−4
14
( −3, 0 ) , ( 3, 0 )
( x − 2 )2 ( y + 3)2
15. (a) Equation:
6
Length of Minor Axis:
+
25
( 0, − 2 10 ) , ( 0, 2 10 )
(exact)
(b) Center:
( 0, − 6.3) , ( 0, 6.3)
(decimal)
(c) Vertices of Major Axis:
( 2, − 3)
( −3, − 3) , ( 7, − 3)
10
Length of Major Axis:
2 10
≈ 0.9
7
( 2, − 7 ) , ( 2, 1)
(d) Vertices of Minor Axis:
(g) Graph:
6
5
(e) Foci:
y
F2
8
( −1, − 3) , ( 5, − 3)
(f) Eccentricity:
e=
3
5
3
4
4
3
(g) Graph:
2
1
−6 −5 −4 −3 −2 −1
−1
C
x
1
2
3
4
5
6
2
−3
F1
−4
−6
F1
y
1
7
−4 −3 −2 −1
−1
−2
−5
=1
16
Length of Minor Axis:
7
7
−3
( 0, − 7 ) , ( 0, 7 )
(d) Vertices of Minor Axis:
e=
6
−2
Length of Major Axis:
(f) Eccentricity:
5
x
−1
(c) Vertices of Major Axis:
(e) Foci:
C
−3 −2 −1
( 0, 0 )
(b) Center:
1
F1
−2
−3
x
1
2
C
5
6
7
8
F2
−4
−7
−5
−8
−6
−7
−8
MATH 1330 Precalculus
229
Odd-Numbered Answers to Exercise Set 8.2: Ellipses
( x + 4 )2 ( y − 3)2
17. (a) Equation:
+
9
1
(g) Graph:
=1
3
( −4, 3)
(b) Center:
y
2
F2
1
( −7, 3) , ( −1, 3)
(c) Vertices of Major Axis:
−3 −2 −1
2
x
3
4
5
6
7
8
−1
−2
6
Length of Major Axis:
1
−3
( −4, 2 ) , ( −4, 4 )
(d) Vertices of Minor Axis:
−5
2
Length of Minor Axis:
C
−4
−6
(
(e) Foci:
) (
−4 − 2, 3 , −4 + 2, 3
)
( −5.4, 3) , ( −2.6, 3)
−7
(exact)
−8
(decimal)
F1
−9
−10
(f) Eccentricity:
2
≈ 0.5
3
e=
−11
(g) Graph:
5
y
C
F1
3
F2
x2 y2
+
=1
9
4
21. (a) Equation:
4
( 0, 0 )
(b) Center:
2
( −3, 0 ) , ( 3, 0 )
(c) Vertices of Major Axis:
1
x
−8
−7
−6
−5
−4
−3
−2
−1
−1
6
Length of Major Axis:
( 0, − 2 ) , ( 0, 2 )
(d) Vertices of Minor Axis:
4
Length of Minor Axis:
(−
(e) Foci:
( x − 2)
19. (a) Equation:
2
+
11
( y + 4)
2
36
(f) Eccentricity:
(c) Vertices of Major Axis:
Length of Major Axis:
( 2, − 10 ) , ( 2, 2 )
12
5, 0
)
(exact)
( −2.2, 0 ) , ( 2.2, 0 )
=1
( 2, − 4 )
(b) Center:
) (
5, 0 ,
e=
(decimal)
5
≈ 0.7
3
(g) Graph:
3
y
(d) Vertices of Minor Axis:
(2 −
) (
11, − 4 , 2 + 11, − 4
( −1.3, − 4 ) , ( 5.3, − 4 )
)
2
(exact)
1
F1
(decimal)
−4
Length of Minor Axis:
(e) Foci:
( 2, − 9 ) , ( 2, 1)
2 11 ≈ 6.6
−3
−2
F2
C
−1
1
2
x
3
4
−1
−2
−3
(f) Eccentricity:
e=
5
6
Continued in the next column…
230
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 8.2: Ellipses
23. (a) Equation:
( x − 1)2 ( y + 2 )2
+
16
25
(g) Graph:
=1
y
9
(1, − 2 )
(b) Center:
8
(c) Vertices of Major Axis:
Length of Major Axis:
(d) Vertices of Minor Axis:
Length of Minor Axis:
(1, − 7 ) , (1, 3)
7
10
6
( −3, − 2 ) , ( 5, − 2 )
5
8
4
(1, − 5 ) , (1, 1)
(e) Foci:
F2
C
3
F1
2
3
e=
5
(f) Eccentricity:
1
x
(g) Graph:
−3
4
−2
−1
1
2
3
4
5
y
3
2
F2
1
−6 −5 −4 −3 −2 −1
−1
1
x
2
3
4
5
6
7
+
7
8
C
−2
( x + 2 )2 ( y − 3)2
27. (a) Equation:
=1
16
( −2, 3)
(b) Center:
−3
(c) Vertices of Major Axis:
−4
F1
−5
8
Length of Major Axis:
−6
−7
( −2, − 1) , ( −2, 7 )
(d) Vertices of Minor Axis:
−8
( −2 −
) (
7, 3 , −2 + 7, 3
)
(exact)
( −4.6, 3) , ( 0.6, 3)
25. (a) Equation:
(b) Center:
( x − 1)2 ( y + 5)2
4
+
16
(g) Graph:
y
( −1, 5) , ( 3, 5 )
) (
(e) Foci: 1, 5 − 2 3 , 1, 5 + 2 3
)
(1, 8.5 ) , (1, 1.5 )
e=
3
4
7
4
Length of Minor Axis:
(f) Eccentricity:
e=
(1, 1) , (1, 9 )
8
(d) Vertices of Minor Axis:
(
( −2, 0 ) , ( −2, 6 )
(e) Foci:
(f) Eccentricity:
Length of Major Axis:
2 7 ≈ 5.3
Length of Minor Axis:
=1
(1, 5 )
(c) Vertices of Major Axis:
(decimal)
6
F2
(exact)
5
(decimal)
4
3
≈ 0.9
2
3
C
2
1
F1
Continued in the next column…
−6
−5
−4
−3
−2
x
−1
1
2
−1
−2
MATH 1330 Precalculus
231
Odd-Numbered Answers to Exercise Set 8.2: Ellipses
29.
31.
x2 y2
+
=1
64 25
( x + 4 )2 ( y − 7 )2
+
9
33.
35.
+
( x − 2 )2
56
37.
11
( x − 3 )2 ( y − 2 )2
+
9
( x + 5 )2 ( y − 2 )2
+
28
=1
64
(b) Center:
( 0, 0 )
(c) Radius:
(d) Graph:
6
7
6
5
4
3
2
1
=1
y
x
C
−7 −6 −5 −4 −3 −2 −1
−1
−2
−3
−4
−5
−6
−7
y2
=1
81
+
25
41.
+
4
( x + 3)2 ( y − 1)2
36
39.
25
( x + 3 )2 ( y + 5 )2
9
x 2 + y 2 = 36
61. (a) Equation:
1 2 3 4 5 6 7
=1
=1
=1
( x − 3)2 + ( y + 2 )2 = 16
( 3, − 2 )
63. (a) Equation:
(b) Center:
(c) Radius:
(d) Graph:
4
3
y
2
2
x2 ( y − 6)
43.
+
=1
32
36
1
x
−2
−1
1
2
3
4
5
6
7
8
−1
45.
( x − 4)
2
25
+
( y − 3)
2
21
−2
=1
C
−3
−4
2
47.
2
x
y
+
=1
16 7
−5
−6
−7
2
2
49. x + y = 81
51.
( x − 7 )2 + ( y + 2 )2 = 100
53.
( x + 3)2 + ( y + 4 )2 = 18
55.
( x − 2 )2 + ( y + 5 )2 = 26
65. (a) Equation:
(b) Center:
(c) Radius:
(d) Graph:
( x + 5 ) 2 + ( y − 2 )2 = 4
( −5, 2 )
2
4
57.
59.
( x + 3)2 + ( y + 3)2 = 10
( x + 3)
2
y
3
2
C
2
+ ( y − 5 ) = 25
1
x
−7
−6
−5
−4
−3
−2
−1
−1
232
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 8.2: Ellipses
( x + 4 )2 + ( y + 3)2 = 12
( −4, − 3)
67. (a) Equation:
(b) Center:
(c) Radius:
(d) Graph:
73. (a) Equation:
(b) Center:
(c) Radius:
(d) Graph:
2 3 ≈ 3.5
1
( x + 3)2 + ( y − 4 )2 = 4
( −3, 4 )
2
y
yx
−8 −7 −6 −5 −4 −3 −2 −1
6
1
−1
5
−2
4
C
−3
C
−4
3
−5
2
−6
1
−7
x
−6
−5
−4
−3
−2
−1
1
−1
2
( x + 1) + ( y − 5)
( −1, 5)
69. (a) Equation:
(b) Center:
(c) Radius:
(d) Graph:
2
=9
75.
( x − 3)2 + ( y − 2 )2 = 32
77.
( x − 5 )2 + ( y + 1)2 = 13
3
9
y
8
7
6
C
5
4
3
2
1
x
−5
−4 −3
−2
−1
1
2
3
4
−1
( x + 5 ) 2 + ( y − 4 )2 = 5
( −5, 4 )
71. (a) Equation:
(b) Center:
(c) Radius:
(d) Graph:
5 ≈ 2.2
7
y
6
5
4
C
3
2
1
x
−8
−7
−6
−5
−4
−3
−2
−1
−1
MATH 1330 Precalculus
233
Odd-Numbered Answers to Exercise Set 8.3: Hyperbolas
1.
Parabola
(f) Asymptotes:
3
y=± x
7
e=
3.
Hyperbola
5.
Circle
(g) Eccentricity:
7.
Ellipse
(h) Graph:
9.
11.
y2 x2
−
=1
8
1
( x − 3)2 ( y − 1)2
5
13.
58
≈ 2.5
3
−
=1
5
( x + 1)2 ( y − 2 )2
5
−
=1
7
15. 2a
17. (a) y − y1 = m ( x − x1 )
23. (a) Equation:
(b) y − k = m ( x − h )
(c) Rise:
Run:
(d) Slopes:
±a
±b
a
a
a
, − ; i.e. ±
b
b
b
x2 y2
−
=1
25 9
(b) Center:
( 0, 0 )
(c) Vertices:
( −5, 0 ) , ( 5, 0 )
10
Length of transverse axis:
a
a
( x − h) , y − k = − ( x − h);
b
b
a
i.e. y − k = ± ( x − h )
b
(e) y − k =
( 0, − 3) , ( 0, 3)
(d) Endpoints of conjugate axis:
6
Length of conjugate axis:
(e) Foci:
(−
) (
34, 0 ,
34, 0
)
( −5.8, 0 ) , ( 5.8, 0 )
19. In the standard form for the equation of a hyperbola,
a 2 represents the denominator of the first term.
(f) Asymptotes:
3
y=± x
5
y2 x2
−
=1
9 49
(g) Eccentricity:
e=
(b) Center:
( 0, 0 )
(h) Graph:
(c) Vertices:
( 0, − 3) , ( 0, 3)
21. (a) Equation:
(exact)
(decimal)
34
≈ 1.2
5
Length of transverse axis: 6
( −7, 0 ) , ( 7, 0 )
(d) Endpoints of conjugate axis:
Length of conjugate axis: 14
(e) Foci:
( 0, −
) (
58 , 0, 58
( 0, − 7.6 ) , ( 0, 7.6 )
)
(exact)
(decimal)
Continued in the next column…
234
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 8.3: Hyperbolas
25. (a) Equation:
−
16
=1
9
(b) Center:
( −1, 5)
(c) Vertices:
( −5, 5) , ( 3, 5 )
8
Length of transverse axis:
( −1, 2 ) , ( −1, 8 )
(d) Endpoints of conjugate axis:
6
Length of conjugate axis:
(e) Foci:
(h) Graph:
( x + 1)2 ( y − 5 )2
( −6, 5 ) , ( 4, 5 )
(f) Asymptotes:
y −5 = ±
(g) Eccentricity:
e=
3
( x + 1)
4
5
4
29. (a) Equation:
(h) Graph:
( x + 4 )2 ( y + 3)2
−
25
1
(b) Center:
( −4, − 3)
(c) Vertices:
( −9, − 3) , (1, − 3)
=1
10
Length of transverse axis:
(d) Endpoints of conjugate axis: ( −4, − 4 ) , ( −4, − 2 )
2
Length of conjugate axis:
(e) Foci:
( −4 −
)(
26, − 3 , −4 + 26, − 3
( −9.1, − 3) , (1.1, − 3)
27. (a) Equation:
( y − 3)
2
25
−
( x − 1)
(1, 3)
(c) Vertices:
(1, − 2 ) , (1, 8 )
(d) Endpoints of conjugate axis:
61
e=
(decimal)
1
( x + 4)
5
26
≈ 1.0
5
(h) Graph:
( −5, 3) , ( 7, 3)
12
Length of conjugate axis:
(1, 3 ±
(g) Eccentricity:
(exact)
10
Length of transverse axis:
(e) Foci:
y+3 = ±
=1
36
(b) Center:
(f) Asymptotes:
2
)
)
(exact)
(1, − 4.8 ) , (1, 10.8 )
(decimal)
5
( x − 1)
6
(f) Asymptotes:
y −3 = ±
(g) Eccentricity:
61
e=
≈ 1.6
5
Note: The foci are very close to the vertices in
the above diagram.
Continued in the next column…
MATH 1330 Precalculus
235
Odd-Numbered Answers to Exercise Set 8.3: Hyperbolas
( x + 1)2 ( y − 1)2
31. (a) Equation:
−
9
64
(b) Center:
( −1, 1)
(c) Vertices:
( −4, 1) , ( 2, 1)
(g) Eccentricity:
6
Length of transverse axis:
(d) Endpoints of conjugate axis:
( −1, − 7 ) , ( −1, 9 )
( −1 −
) (
)
73, 1 , −1 + 73, 1
( −9.5, 1) , ( 7.5, 1)
(f) Asymptotes:
y −1 = ±
(g) Eccentricity:
e=
e=
41
≈ 3.2
2
(h) Graph:
16
Length of conjugate axis:
(e) Foci:
2 5
( x + 3) (exact)
5
y − 5 ≈ ±0.9 ( x + 3)
(decimal)
(f) Asymptotes: y − 5 = ±
=1
(exact)
(decimal)
8
( x + 1)
3
73
≈ 2.8
3
(h) Graph:
Numbers 35-57 can be found on the next page…
33. (a) Equation:
( y − 5 )2 ( x + 3)2
−
4
5
(b) Center:
( −3, 5 )
(c) Vertices:
( −3, 3) , ( −3, 7 )
=1
4
Length of transverse axis:
(d) Endpoints of conjugate axis:
( −3 −
) (
5, 5 , −3 + 5, 5
)
( −5.2, 5 ) , ( −0.7, 5 )
(exact)
(decimal)
Length of conjugate axis:
2 5 ≈ 4.5
( −3, 5 −
)
(e) Foci:
) (
41 , −3, 5 + 41
( −3, − 1.4 ) , ( −3, 11.4 )
(exact)
(decimal)
Continued in the next column…
236
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set 8.3: Hyperbolas
35. (a) Equation:
( x − 9 )2
−
108
(b) Center:
( 9, 0 )
(c) Vertices:
(9 − 6
y2
=1
36
49.
9
) (
3, 0 , 9 + 6 3, 0
)
(exact)
51.
Length of transverse axis:
(d) Endpoints of conjugate axis:
Length of conjugate axis:
12 3 ≈ 20.8
( 9, − 6 ) , ( 9, 6 )
3
( x − 9 ) (exact)
3
y ≈ 0.6 ( x − 9 )
(decimal)
55.
(g) Eccentricity:
e=
57.
−
49
−
128
( y + 3)2 ( x − 4 )2
36
y=±
16
( x − 3 )2 ( y − 4 )2
16
12
( −3, 0 ) , ( 21, 0 )
(f) Asymptotes:
53.
−
( x + 1)2 ( y − 2 )2
16
( −1.4, 0 ) , (19.4, 0 )
(e) Foci:
( y − 3 )2 ( x − 5 )2
−
45
( y − 4 )2 ( x − 3 )2
9
−
7
=1
=1
=1
=1
=1
2 3
≈ 1.1
3
(h) Graph:
37.
39.
x2 y2
−
=1
64 25
( y + 5 )2 ( x + 2 )2
4
41.
45.
−
16
=1
=1
y2 x2
−
=1
9 72
( x − 4 )2 ( y − 3 )2
11
47.
100
( y − 1)2 ( x + 6 )2
25
43.
−
−
25
( y − 1)2 ( x − 4 )2
64
−
16
=1
=1
MATH 1330 Precalculus
237
Odd-Numbered Answers to Exercise Set A.1:
Factoring Polynomials
1.
(a) 13
(b) No
3.
(a) 100
(b) Yes
5.
(a) 36
(b) Yes
7.
(a) 81
(b) Yes
9.
(a) −36
(b) No
11.
( x + 5 )( x − 1)
13.
( x − 3)( x − 2 )
15. x 2 − 7 x − 12
17.
( x + 10 )( x + 2 )
19.
( x + 3)( x − 8 )
21.
( x + 8 )2
23.
( x − 7 )( x − 8 )
25.
( x + 4 )( x − 15 )
27.
( x + 3)( x + 14 )
29.
( x + 7 )( x − 7 )
51. −5 x 2 ( x + 2 )( x − 2 )
53. 2 ( x + 4 )( x + 1)
55. −10 ( x − 7 )( x + 6 )
57. x ( x + 11)( x − 2 )
59. − x ( x + 2 )
(
2
61. x 2 x 2 + 6 x + 6
)
63. x 3 ( 3 x + 10 )( 3 x − 10 )
65. 5 ( 2 x + 1)( 5 x + 3 )
67.
( x + 2 )( x + 5 )( x − 5)
69.
( x − 5) ( x2 + 4)
71.
( x + 9 )( 2 x + 1)( 2 x − 1)
31. x 2 − 3
33. 9 x 2 + 25
35.
( 2 x + 1)( x − 3)
37.
( 2 x + 1)( 4 x − 3)
39.
( 3x + 4 )( 3x − 1)
41.
( 4 x + 5 )( x − 2 )
43.
( 4 x − 3)( 3x − 2 )
45. x ( x + 9 )
47. −5 x ( x − 4 )
49. 2 ( x + 3)( x − 3)
238
University of Houston Department of Mathematics
Odd-Numbered Answers to Exercise Set A.2
Dividing Polynomials
1.
Quotient:
x−4;
Remainder: 3
3.
Quotient:
x+6;
Remainder: −8
35.
x 2 − 11x + 24 = ( x − 8 )( x − 3)
37. x 2 − 7 x − 18 = ( x + 2 )( x − 9 )
39. 4 x 2 − 25 x − 21 = ( x − 7 )( 4 x + 3)
5.
Quotient:
x2 − 5x − 4 ;
Remainder: 0
7.
Quotient:
3x 2 + 4 x + 5 ;
Remainder: −7
9.
Quotient:
2x + 7 ;
Remainder: 5 x + 14
11. Quotient:
1 x3
2
41. 2 x 2 + 7 x + 5 = ( x + 1)( 2 x + 5 )
+ 8x2 − 1 ;
Remainder: −8
13. Quotient:
3 x 2 − x − 15 ;
Remainder: 5 x + 60
15. Quotient:
x+2;
Remainder: 24
17. Quotient:
3x 2 − 2 x + 4 ;
Remainder: 8
19. Quotient:
x3 − x 2 + 4 x − 4 ;
Remainder: 0
21. Quotient:
3 x3 + 4 x 2 − 7 x − 17 ;
Remainder: −75
23. Quotient:
x2 − 2x + 4 ;
Remainder: 0
25. Quotient:
4x2 + 2x − 6 ;
Remainder: 2
27. (a) Using substitution, P(2) = 4
(b) The remainder is 4 , so P(2) = 4 .
29. (a) P(−1) = −7
(b) The remainder is −7 , so P(−1) = −7 .
31. P(5) = −97
( )
33. P − 34 = −10
MATH 1330 Precalculus
239
